JEE Main
Mathematics
Matrices and Determinants
Previous Years Questions

For the system of linear equations $$\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta$$, which one of the following statements is NOT correct?...
If $$A = {1 \over 2}\left[ {\matrix{ 1 & {\sqrt 3 } \cr { - \sqrt 3 } & 1 \cr } } \right]$$, then :
Let $$S$$ denote the set of all real values of $$\lambda$$ such that the system of equations $$\lambda x+y+z=1$$ $$x+\lambda y+z=1$$ $$x+y+\lambda z=1... For the system of linear equations$$x+y+z=6\alpha x+\beta y+7 z=3x+2 y+3 z=14$$which of the following is NOT true ?... Let$$A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 1} \cr 0 & {12} & { - 3} \cr } } \right)$$. Then the sum of the diagonal element... For \alpha, \beta \in \mathbb{R}, suppose the system of linear equations$$ \begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \...
If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>1$, then :
Let the system of linear equations $$x+y+kz=2$$ $$2x+3y-z=1$$ $$3x+4y+2z=k$$ have infinitely many solutions. Then the system $$(k+1)x+(2k-1)y=7$$ $$(2... Let$$A=\left(\begin{array}{cc}\mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q}\end{array}\right), \mathrm{d}=|\mathrm{A}| \neq 0$$and$$\mathrm{|A...
The set of all values of $$\mathrm{t\in \mathbb{R}}$$, for which the matrix $$\left[ {\matrix{ {{e^t}} & {{e^{ - t}}(\sin t - 2\cos t)} & {{e^{ - t... Let$$\alpha$$and$$\beta$$be real numbers. Consider a 3$$\times$$3 matrix A such that$$A^2=3A+\alpha I$$. If$$A^4=21A+\beta I$$, then Consider the following system of equations$$\alpha x+2y+z=12\alpha x+3y+z=13x+\alpha y+2z=\beta$$for some$$\alpha,\beta\in \mathbb{R}$$. ... Let A, B, C be 3$$\times$$3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements (S1) A$$^{13}$$B$$^{26}-...
Let $$A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}... Let$$x,y,z > 1$$and$$A = \left[ {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 2 & {{{\log }_y}z} \cr {{{\log }_z}...
Let S$$_1$$ and S$$_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{ 0\}$$ for which the system of linear equations $$ax + 2ay - 3az = 1$$...
Let A be a 3 $$\times$$ 3 matrix such that $$\mathrm{|adj(adj(adj~A))|=12^4}$$. Then $$\mathrm{|A^{-1}~adj~A|}$$ is equal to
If the system of equations $$x+2y+3z=3$$ $$4x+3y-4z=4$$ $$8x+4y-\lambda z=9+\mu$$ has infinitely many solutions, then the ordered pair ($$\lambda,\mu... The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all t... If A and B are two non-zero n$$\times$$n matrices such that$$\mathrm{A^2+B=A^2B}$$, then Let$$\alpha$$be a root of the equation$$(a - c){x^2} + (b - a)x + (c - b) = 0$$where a, b, c are distinct real numbers such that the matrix$$\lef...
Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementa...
If the system of equations \begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned} has infinitely many solutions, the...
Let A and B be two $$3 \times 3$$ non-zero real matrices such that AB is a zero matrix. Then
Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT...
Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix $$B_{0}=A^{49}+2 A^{98}$$. If $$B_{n... Let$$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$. If$$\mathrm{A}^{2}+\gamma \mathrm{A}+18 \mathrm{I}=\mathrm{O}$$, then$$...
Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let $$\alpha, \beta \in \mathbb{R}$$ be such that $$\alpha A^{2}+\beta A=2 I$$. T...
$$\text { Let } A=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} 9^{2} & -10^{2} & 11^{2} \\ 12^{2} &... If the system of linear equations.$$8x + y + 4z = - 2x + y + z = 0\lambda x - 3y = \mu $$has infinitely many solutions, then the distance... Let A be a 2$$\times$$2 matrix with det (A) =$$-$$1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :... The number of real values of$$\lambda$$, such that the system of linear equations 2x$$-$$3y + 5z = 9 x + 3y$$-$$z =$$-$$18 3x$$-$$y + ($$\lamb...
The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations \begin{aligned} &3(\sin 3 \theta) x-y+z=2 \\ &3(\cos 2 \theta) x+4 ... LetA = \left[ {\matrix{ 1 & { - 2} & \alpha \cr \alpha & 2 & { - 1} \cr } } \right]$$and$$B = \left[ {\matrix{ 2 & \alpha \cr ...
Let A and B be two square matrices of order 2. If $$det\,(A) = 2$$, $$det\,(B) = 3$$ and $$\det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}... Let$$A = \left( {\matrix{ 2 & { - 1} \cr 0 & 2 \cr } } \right)$$. If$$B = I - {}^5{C_1}(adj\,A) + {}^5{C_2}{(adj\,A)^2} - \,\,.....\,\, ...
If the system of linear equations 2x + y $$-$$ z = 7 x $$-$$ 3y + 2z = 1 x + 4y + $$\delta$$z = k, where $$\delta$$, k $$\in$$ R has infinitely many s...
Let $$A = [{a_{ij}}]$$ be a square matrix of order 3 such that $${a_{ij}} = {2^{j - i}}$$, for all i, j = 1, 2, 3. Then, the matrix A2 + A3 + ...... +...
If the system of linear equations $$2x + 3y - z = - 2$$ $$x + y + z = 4$$ $$x - y + |\lambda |z = 4\lambda - 4$$ where, $$\lambda$$ $$\in$$ R, has n...
Let A be a matrix of order 3 $$\times$$ 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________.
Let $$f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R$$. Then the sum ...
Let A and B be two 3 $$\times$$ 3 matrices such that $$AB = I$$ and $$|A| = {1 \over 8}$$. Then $$|adj\,(B\,adj(2A))|$$ is equal to
Let the system of linear equations $$x + 2y + z = 2$$, $$\alpha x + 3y - z = \alpha$$, $$- \alpha x + y + 2z = - \alpha$$ be inconsistent. Then $$... If the system of equations$$\alpha$$x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z =$$\beta$$has infinitely many solutions, then the ordered pair ($$\...
Let A be a 3 $$\times$$ 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to :
The ordered pair (a, b), for which the system of linear equations 3x $$-$$ 2y + z = b 5x $$-$$ 8y + 9z = 3 2x + y + az = $$-$$1 has no solution, is :...
The system of equations $$- kx + 3y - 14z = 25$$ $$- 15x + 4y - kz = 3$$ $$- 4x + y + 3z = 4$$ is consistent for all k in the set...
Let A be a 3 $$\times$$ 3 real matrix such that $$A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \left( {\matrix{ 1 \cr 1 ... Let$$A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$$. If M and N are two matrices given by$$M = \sum\limits_{k = 1}^{10} {{A...
Let the system of linear equations x + y + $$\alpha$$z = 2 3x + y + z = 4 x + 2z = 1 have a unique solution (x$$^ *$$, y$$^ *$$, z$$^ *$$). If ($$\... The number of values of$$\alpha$$for which the system of equations : x + y + z =$$\alpha\alpha$$x + 2$$\alpha$$y + 3z =$$-$$1 x + 3$$\alpha$$... Let S = {$$\sqrt{n}$$: 1$$\le$$n$$\le$$50 and n is odd}. Let a$$\in$$S and$$A = \left[ {\matrix{ 1 & 0 & a \cr { - 1} & 1 & 0 \cr ...
Consider the system of linear equations$$-$$x + y + 2z = 03x $$-$$ ay + 5z = 12x $$-$$ 2y $$-$$ az = 7Let S1 be the set of all a$$\in$$R for which the...
Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}$$, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A = {[{... If$$\alpha$$+$$\beta$$+$$\gamma$$= 2$$\pi$$, then the system of equations x + (cos$$\gamma$$)y + (cos$$\beta$$)z = 0(cos$$\gamma$$)x + y + (c... If the following system of linear equations2x + y + z = 5x$$-$$y + z = 3x + y + az = bhas no solution, then : If$${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i =$$\sqrt { - 1} $$, then the determinant$$\left| {\matrix{ ...
Let $$A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} ... Let A(a, 0), B(b, 2b + 1) and C(0, b), b$$\ne$$0, |b|$$\ne$$1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all pos... Let [$$\lambda$$] be the greatest integer less than or equal to$$\lambda$$. The set of all values of$$\lambda$$for which the system of linear equat... If the matrix$$A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)$$satisfies$$A({A^3} + 3I) = 2I$$, then the value of K ... Let$$A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 0 \cr } } \right)$$. Then A2025$$-$$A2020 is... Let$$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations$$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\thet...
If $$A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \c... Let A and B be two 3$$\times$$3 real matrices such that (A2$$-$$B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determ... Let$$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$$. If A$$-$$1 =$$\alpha$$I +$$\beta$$A,$$\alpha$$,$$\beta...
The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr... If$$P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]$$, then P50 is : The values of a and b, for which the system of equations 2x + 3y + 6z = 8x + 2y + az = 53x + 5y + 9z = bhas no solution, are :... The values of$$\lambda$$and$$\mu$$such that the system of equations$$x + y + z = 6$$,$$3x + 5y + 5z = 26$$,$$x + 2y + \lambda z = \mu $$has no... Let A = [aij] be a real matrix of order 3$$\times$$3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix... The value of k$$\in$$R, for which the following system of linear equations3x$$-$$y + 4z = 3,x + 2y$$-$$3z =$$-$$26x + 5y + kz =$$-$$3,has infin... Let$$A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$$, a$$\in$$R be written as P + Q where P is a symmetric matrix and Q is... Let the system of linear equations 4x +$$\lambda$$y + 2z = 02x$$-$$y + z = 0$$\mu$$x + 2y + 3z = 0,$$\lambda$$,$$\mu\in$$R.has a non-trivial ... Define a relation R over a class of n$$\times$$n real matrices A and B as "ARB iff there exists a non-singular matrix P such that PAP$$-$$1 = B". Th... The solutions of the equation$$\left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr {{{\cos }^2}x} & {1 + {{...
Let $$A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \cr } } \right]$$ and $$2A - ... If x, y, z are in arithmetic progression with common difference d, x$$\ne$$3d, and the determinant of the matrix$$\left[ {\matrix{ 3 & {4\sq...
The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to :
If $$A = \left( {\matrix{ 0 & {\sin \alpha } \cr {\sin \alpha } & 0 \cr } } \right)$$ and $$\det \left( {{A^2} - {1 \over 2}I} \ri... Let$$A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1} $$. Then, the system of linear equations$${...
Consider the following system of equations :x + 2y $$-$$ 3z = a2x + 6y $$-$$ 11z = bx $$-$$ 2y + 7z = c,where a, b and c are real constants. Then the ...
Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices...
The value of $$\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & 1 \cr {(a + 3)(a + 4)} &a... Let A be a 3$$\times$$3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2$$ \to $$2... If for the matrix,$$A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right]$$,$$A{A^T} = {I_2}$$, then the va... The following system of linear equations2x + 3y + 2z = 93x + 2y + 2z = 9x$$-$$y + 4z = 8 Let A and B be 3$$\times$$3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 ... For the system of linear equations:$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,consider the following statements :(A) The system has unique s... The system of linear equations 3x - 2y - kz = 10 2x - 4y - 2z = 6 x+2y - z = 5m is inconsistent if : Let$$\theta = {\pi \over 5}$$and$$A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \theta } & {\cos \theta } \c...
Let m and M be respectively the minimum and maximum values of $$\left| {\matrix{ {{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr {... The values of$$\lambda $$and$$\mu $$for which the system of linear equations x + y + z = 2 x + 2y + 3z = 5 x + 3y +$$\lambda $$z =$$\mu $$has i... If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then$$\left| {\matrix{ x & {a + y} & {x + a} \cr...
If the system of linear equations x + y + 3z = 0 x + 3y + k2z = 0 3x + y + 3z = 0 has a non-zero solution (x, y, z) for some k $$\in$$ R, then x + $... Let $$\lambda \in$$ R . The system of linear equations 2x1 - 4x2 + $$\lambda$$x3 = 1 x1 - 6x2 + x3 = 2 $$\lambda$$x1 - 10x2 + 4x3 = 3 is inconsist... Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 a... If the system of equations x+y+z=2 2x+4y–z=6 3x+2y+$$\lambda$$z=$$\mu$$ has infinitely many solutions, then If $$A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr } } \right]$$, $$\left( {\thet... Let A be a 3$$ \times $$3 matrix such that adj A =$$\left[ {\matrix{ 2 & { - 1} & 1 \cr { - 1} & 0 & 2 \cr 1 & { -... If $$\Delta$$ = $$\left| {\matrix{ {x - 2} & {2x - 3} & {3x - 4} \cr {2x - 3} & {3x - 4} & {4x - 5} \cr {3x - 5} & {... Let a, b, c$$ \in $$R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A =$$\left( {\matrix{ a & b & c \cr b & c &am... Let A = {X = (x, y, z)T: PX = 0 and x2 + y2 + z2 = 1} where $$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr ... Let S be the set of all$$\lambda  \in $$R for which the system of linear equations 2x – y + 2z = 2 x – 2y +$$\lambda $$z = –4 x +$$\lambda $$... Let A be a 2$$ \times $$2 real matrix with entries from {0, 1} and |A|$$ \ne $$0. Consider the following two statements : (P) If A$$ \ne $$I2 , ... The following system of linear equations 7x + 6y – 2z = 0 3x + 4y + 2z = 0 x – 2y – 6z = 0, has If the matrices A =$$\left[ {\matrix{ 1 & 1 & 2 \cr 1 & 3 & 4 \cr 1 & { - 1} & 3 \cr } } \right]$$, B = adjA... If for some$$\alpha $$and$$\beta $$in R, the intersection of the following three places x + 4y – 2z = 1 x + 7y – 5z = b x + 5y + az = 5 is a line ... The system of linear equations$$\lambda $$x + 2y + 2z = 5 2$$\lambda $$x + 3y + 5z = 8 4x +$$\lambda $$y + 6z = 10 has If$$A = \left( {\matrix{ 2 & 2 \cr 9 & 4 \cr } } \right)$$and$$I = \left( {\matrix{ 1 & 0 \cr 0 & 1 \cr } }... For which of the following ordered pairs ($$\mu$$, $$\delta$$), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = $$\mu$$ 4x + 4y + 4z ... Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3. If the determinant of B is 81, then the de... Let $$\alpha$$ be a root of the equation x2 + x + 1 = 0 and the matrix A = $${1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & 1 & 1 \cr 1 &am... If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c$$ \in $$R are non-zero distinct; has a non-ze... A value of$$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which$$\left| {\matrix{ {1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } &... If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = $$\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \rig... If$$B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1} \cr } } \right]$$is the ... Let$$\lambda $$be a real number for which the system of linear equations x + y + z = 6, 4x +$$\lambda $$y –$$\lambda $$z =$$\lambda $$– 2, 3x + ... The sum of the real roots of the equation$$\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} & {x - 3} \cr { - 3} &am... If $${\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - x} & 1 \cr {\cos \the... If the system of linear equations x + y + z = 5 x + 2y + 2z = 6 x + 3y +$$\lambda $$z =$$\mu $$, ($$\lambda $$,$$\mu  \in $$R), has infinitel... If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution (x, y, z), then$${x \over y} + {y \over z} +... The total number of matrices $$A = \left( {\matrix{ 0 & {2y} & 1 \cr {2x} & y & { - 1} \cr {2x} & { - y} & 1 \cr... Let$$\alpha $$and$$\beta $$be the roots of the equation x2 + x + 1 = 0. Then for y$$ \ne $$0 in R,$$$\left| {\matrix{ {y + 1} & \alpha ...
If $$\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]$$$$\left... Let the number 2,b,c be in an A.P. and A =$$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} ...
Let $$A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr } } \right)$$, ($$\alpha$$ ...
The greatest value of c $$\in$$ R for which the system of linear equations x – cy – cz = 0 cx – y + cz = 0 cx + cy – z = 0 has a non-trivial solutio...
If   A = $$\left[ {\matrix{ 1 & {\sin \theta } & 1 \cr { - \sin \theta } & 1 & {\sin \theta } \cr { - 1} & ... The set of all values of$$\lambda $$for which the system of linear equations x – 2y – 2z =$$\lambda $$x x + 2y + z =$$\lambda $$y – x – y =$$\lam...
An ordered pair ($$\alpha$$, $$\beta$$) for which the system of linear equations (1 + $$\alpha$$) x + $$\beta$$y + z = 2 $$\alpha$$x + (1 + $$\... Let P =$$\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right]$$and Q = [qij] be two 3$$ ...
If  $$\left| {\matrix{ {a - b - c} & {2a} & {2a} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a... Let A and B be two invertible matrices of order 3$$ \times $$3. If det(ABAT) = 8 and det(AB–1) = 8, then det (BA–1 BT) is equal to : ... If the system of linear equations 2x + 2y + 3z = a 3x – y + 5z = b x – 3y + 2z = c where a, b, c are non zero real numbers, has more one solution, the... Let A =$$\left( {\matrix{ 0 & {2q} & r \cr p & q & { - r} \cr p & { - q} & r \cr } } \right).$$... Let A =$$\left[ {\matrix{ 2 & b & 1 \cr b & {{b^2} + 1} & b \cr 1 & b & 2 \cr } } \right]$$where b > 0... The number of values of$$\theta  \in $$(0,$$\pi $$) for which the system of linear equations x + 3y + 7z = 0$$-$$x + 4y + 7z = 0 (sin3$$\the...
If the system of equations x + y + z = 5 x + 2y + 3z = 9 x + 3y + az = $$\beta$$ has infinitely many solutions, then $$\beta$$ $$-$$ $$\alpha$$ equ...
Let  d $$\in$$ R, and  $$A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr 1 & {\le... If the system of linear equations x$$-$$4y + 7z = g 3y$$-$$5z = h$$-$$2x + 5y$$-$$9z = k is consiste... If$$A = \left[ {\matrix{ {{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr {{e^t}} & { - {e^{ - t}}\cos t - ...
If $$A = \left[ {\matrix{ {\cos \theta } & { - \sin \theta } \cr {\sin \theta } & {\cos \theta } \cr } } \right]$$, then the matri...
The system of linear equations x + y + z = 2 2x + 3y + 2z = 5 2x + 3y + (a2 – 1) z = a + 1 then
The number of values of k for which the system of linear equations, (k + 2)x + 10y = k kx + (k +3)y = k -1 has no solution, is :
Let A = $$\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]$$ and B = A20. Then the sum ...
If $$\left| {\matrix{ {x - 4} & {2x} & {2x} \cr {2x} & {x - 4} & {2x} \cr {2x} & {2x} & {x - 4} \cr } } \righ... If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then$${{xz} \over {{y^2}}}$$i... Suppose A is any 3$$ \times $$3 nonsingular matrx and ( A$$-$$3I) (A$$-$$5I) = O where I = I3 and O = O3. If$$\alpha $$A +$$\beta $$A-1 = 4I, t... If the system of linear equations x + ay + z = 3 x + 2y + 2z = 6 x + 5y + 3z = b has no solution, then : Let$$A$$be a matrix such that$$A.\left[ {\matrix{ 1 & 2 \cr 0 & 3 \cr } } \right]$$is a scalar matrix and |3A| = 108. Then A2... Let S be the set of all real values of k for which the systemof linear equations x + y + z = 2 2x + y$$-$$z = 3 3x + 2y + kz = 4 has a unique soluti... For two 3 × 3 matrices A and B, let A + B = 2BT and 3A + 2B = I3, where BT is the transpose of B and I3 is 3 × 3 identity matrix. Then :... Let A be any 3$$ \times $$3 invertible matrix. Then which one of the following is not always true ? The number of real values of$$\lambda $$for which the system of linear equations 2x + 4y$$-\lambda $$z = 0 4x +$$\lambda $$y + 2z = 0$$\lamb...
If $$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{ 0 & {\cos x} & { - \sin x} \cr {\sin x} & 0 & {\cos x} \cr... If$$A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$, then adj(3A2 + 12A) is equal to If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solu... Let A be a 3$$ \times $$3 matrix such that A2$$-$$5A + 7I = 0 Statement - I : A$$-$$1 =$${1 \over 7}$$(5I$$-$$A). Statement - II ... If A =$$\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$, then the determinant of the matrix (A2016 − 2... If P =$$\left[ {\matrix{ {{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr { - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr } } \right],A =...
The number of distinct real roots of the equation, $$\left| {\matrix{ {\cos x} & {\sin x} & {\sin x} \cr {\sin x} & {\cos x} &am... If$$A = \left[ {\matrix{ {5a} & { - b} \cr 3 & 2 \cr } } \right]$$and$$A$$adj$$A=A{A^T},$$then$$5a+b$$is equal to : The system of linear equations$$\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y - \lambda z = 0} \cr } $$... The set of all values of$$\lambda $$for which the system of linear equations:$$\matrix{ {2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr {2{x_...
If $$A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]$$ is a matrix satisfying...
If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and $$\left| {\matrix{ 3 & {1 + f\left( 1 \right)} & {1... If$$A$$is a$$3 \times 3$$non-singular matrix such that$$AA'=A'A$$and$$B = {A^{ - 1}}A',$$then$$BB'$$equals: If$$P = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right]$$is the adjoint of a ... Let$$A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right)$$. If$${u_1}$$and$${u_2}$... Let $$P$$ and $$Q$$ be $$3 \times 3$$ matrices $$P \ne Q.$$ If $${P^3} = {Q^3}$$ and $${P^2}Q = {Q^2}P$$ then determinant of $$\left( {{P^2} + {Q^2}}... Let$$A$$and$$B$$be two symmetric matrices of order$$3$$. Statement - 1:$$A(BA)$$and$$(AB)A$$are symmetric matrices. Statement - 2:$$A... The number of values of $$k$$ for which the linear equations $$4x + ky + 2z = 0,kx + 4y + z = 0$$ and $$2x+2y+z=0$$ possess a non-zero solution is ... Let $$A$$ be a $$\,2 \times 2$$ matrix with non-zero entries and let $${A^2} = I,$$ where $$I$$ is $$2 \times 2$$ identity matrix. Define $$Tr$$$$(A... Consider the system of linear equations;$$$\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} = 3} \cr {3{x_1} + 5{x_2}...
The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is
Let $$A$$ be a $$\,2 \times 2$$ matrix Statement - 1 : $$adj\left( {adj\,A} \right) = A$$ Statement - 2 :$$\left| {adj\,A} \right| = \left| A \right|... Let$$a, b, c$$be such that$$b\left( {a + c} \right) \ne 0$$if$$\left| {\matrix{ a & {a + 1} & {a - 1} \cr { - b} & {b + 1} &a...
Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal ent...
Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$ $$y=az+cx,$$ and $$z=bx+ay.$$...
Let $$A$$ be a square matrix all of whose entries are integers. Then which one of the following is true?
If$$D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|$$ for $$x \ne 0,y ... Let$$A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right...
If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that $${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then wh...
Let $$A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)$$ and $$B = \left( {\matrix{ a & 0 \cr 0 & b \cr } ... If$${A^2} - A + 1 = 0$$, then the inverse of$$A$$is The system of equations$$\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr {x + y + \alpha \,z = \al...
If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant $$\Delta \left| {\matrix{ {\log {a_n}} & {\log {a_{n + 1}}} &... If$${a^2} + {b^2} + {c^2} = - 2$$and f$$\left( x \right) = \left| {\matrix{ {1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 +...
Let $$A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right).$$ and $$10$$ $$B... Let$$A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right)$$. The only c... If$${a_1},{a_2},{a_3},.........,{a_n},......$$are in G.P., then the value of the determinant$$\left| {\matrix{ {\log {a_n}} & {\log {a_{n +...
If $$A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]$$ and $${A^2} = \left[ {\matrix{ \alpha & \beta \cr \beta ... If$$1,\omega ,{\omega ^2}$$are the cube roots of unity, then$$\Delta = \left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \c...
If the system of linear equations $$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$ has a non - zero solution, then $$a, b, c$$....
If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then $$\left| {\matrix{ a & b & {ax + b} \cr b & c & {bx ... ## Numerical Let A be a n \times n matrix such that |\mathrm{A}|=2. If the determinant of the matrix \operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(... Let A be a symmetric matrix such that$$\mathrm{|A|=2}$$and$$\left[ {\matrix{ 2 & 1 \cr 3 & {{3 \over 2}} \cr } } \right]A = \left[ {\ma...
Let $$\mathrm{A_1,A_2,A_3}$$ be the three A.P. with the same common difference d and having their first terms as $$\mathrm{A,A+1,A+2}$$, respectively....
Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]... The number of matrices of order$$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________.... Let p and p + 2 be prime numbers and let$$ \Delta=\left|\begin{array}{ccc} \mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\ (\mathrm{p}+1) ! & (...
Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \... Consider a matrix$$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+...
Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1,0,1\}$$. The total number of matrices $$A \in S$$ such that the s...
The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such ...
Let $$A=\left[\begin{array}{lll} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right], a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}, A^{n}=... Let$$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$and$$B=A-I$$. If$$\omega=\frac{\sqrt{3} i-1}{2}$$, then ... Let$$M = \left[ {\matrix{ 0 & { - \alpha } \cr \alpha & 0 \cr } } \right]$$, where$$\alpha$$is a non-zero real number an$$N = \sum\li...
If the system of linear equations $$2x - 3y = \gamma + 5$$, $$\alpha x + 5y = \beta + 1$$, where $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R has inf...
Let $$A = \left( {\matrix{ {1 + i} & 1 \cr { - i} & 0 \cr } } \right)$$ where $$i = \sqrt { - 1}$$. Then, the number of elements in the s...
Let A be a matrix of order 2 $$\times$$ 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, ...
The positive value of the determinant of the matrix A, whose Adj(Adj(A)) = $$\left( {\matrix{ {14} & {28} & { - 14} \cr { - 14} & {14} & {28} ... Let$$X = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$and$$Z = ...
Let $$A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$$ and $$B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr ... Let A be a 3$$\times$$3 matrix having entries from the set {$$-$$1, 0, 1}. The number of all such matrices A having sum of all the entries equal to ... Let$$S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right\}$$and let$${T_n} = \{ A \i...
The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{... If the system of linear equations2x + y$$-$$z = 3x$$-$$y$$-$$z =$$\alpha$$3x + 3y +$$\beta$$z = 3has infinitely many solution, then$$\alpha$$... Let A be a 3$$\times$$3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________. If$$A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]$$and M = A + A2 + A3 + ........ For real numbers$$\alpha$$and$$\beta$$, consider the following system of linear equations :x + y$$-$$z = 2, x + 2y +$$\alpha$$z = 1, 2x$$-$$y ... Let$$f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\c...
Let $$M = \left\{ {A = \left( {\matrix{ a & b \cr c & d \cr } } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}$$. Defi...
Let $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]$$. Then the number of 3 $$\t... Let$$A = \{ {a_{ij}}\} $$be a 3$$\times$$3 matrix, where$${a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr ...
Let $$A = \left( {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr 0 & 0 & 1 \cr } } \right)$$ and B = 7A20 $$-... Let a, b, c, d in arithmetic progression with common difference$$\lambda$$. If$$\left| {\matrix{ {x + a - c} & {x + b} & {x + a} \cr ...
Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3} \cr } } \right]$$. Then t...
Let $$A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]$$ and $$B = \left[ {\matrix{ \alpha \cr \beta \cr } } \ri... If 1, log10(4x$$-$$2) and log10$$\left( {{4^x} + {{18} \over 5}} \right)$$are in arithmetic progression for a real number x, then the value of the ... If$$A = \left[ {\matrix{ 2 & 3 \cr 0 & { - 1} \cr } } \right]$$, then the value of det(A4) + det(A10$$-$$(Adj(2A))10) is equal ... Let$$A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]$$and$$B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right...
Let $$P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \ri... The total number of 3$$\times$$3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is e... If the matrix$$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]$$satisfies t... If$$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr }...
Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$, where x, y and z are real...
If the system of equationskx + y + 2z = 13x $$-$$ y $$-$$ 2z = 2$$-$$2x $$-$$2y $$-$$4z = 3has infinitely many solutions, then k is equal to _________...
Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, wher...
Let M be any 3 $$\times$$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements ...
The sum of distinct values of $$\lambda$$ for which the system of equations$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lam... If the system of equations x - 2y + 3z = 9 2x + y + z = b x - 7y + az = 24, has infinitely many solutions, then a - b is equal to............ Let S be the set of all integer solutions, (x, y, z), of the system of equations x – 2y + 5z = 0 –2x + 4y + z = 0 –7x + 14y + 9z = 0 such that 15$$ \...
Let A = $$\left[ {\matrix{ x & 1 \cr 1 & 0 \cr } } \right]$$, x $$\in$$ R and A4 = [aij]. If a11 = 109, then a22 is equal to ___...
The number of all 3 × 3 matrices A, with enteries from the set {–1, 0, 1} such that the sum of the diagonal elements of AAT is 3, is
If the system of linear equations, x + y + z = 6 x + 2y + 3z = 10 3x + 2y + $$\lambda$$z = $$\mu$$ has more than two solutions, then $$\mu$$ - \l...
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