Matrices and Determinants · Mathematics · TS EAMCET

A is a $3 \times 3$ matrix satisfying $A^3-5 A^2+7 A+I=0$ If $A^5-6 A^4+12 A^3-6 A^2+2 A+2 I=l A+m I$, then $l+m=$

If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1\end{array}\right], A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 y \\ 5 & -3 & 1\end{array}\right]$, then the point $(x, y)$ lies on the curve represented by the equation.

Consider a homogeneous system of three linear equations in three unknowns represented by $A X=0$.

If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right], l \neq 0, m \neq 0, l, m \in R$ represents an infinite number of solutions of this system, then rank of $A$ is

The number of real values of ' $a$ ' for which the system of equations $2 x+3 y+a z=0, x+a y-2 z=0$ and $3 x+y+3 z=0$ has non-trivial solution is

If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2 x+3 y+z=-1,3 x+y+z=4$, $x-3 y-2 z=1$, then the value of $\beta$ is

The positive value of ' $a$ ' for which the system of linear homogeneous equations $x+a y+z=0, a x+2 y-z=0$, $2 x+3 y+z=0$ has non-trivial solution is

If $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1\end{array}\right]$ then $|\operatorname{adj}|\left(A^2\right) \mid=$

If the system of simultaneous linear equations $x-2 y+z=0,2 x+3 y+z=6$ and $x+2 y+p z=q$ has infinitely many solutions, then

If the system of linear equations $(\sin \theta) x-y+z=0$, $x-(\cos \theta) y+z=0, x+y+(\sin \theta) z=0$ has non-trivial solution, then the least positive value of $\theta$ is

6. If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 1 \\ 1 & 3 & 1\end{array}\right]$ and $B=\left[\begin{array}{lll}2 & 3 & 4 \\ 3 & 2 & 2 \\ 2 & 4 & 2\end{array}\right]$, then $\sqrt{|\operatorname{adj}(A B)|}=$

7. If $A=\left[\begin{array}{lll}1 & 5 & 2 \\ 4 & 1 & 3 \\ 2 & 6 & 3\end{array}\right]$, then $\left|(\operatorname{adj} A)^{-1}\right|=$

If the system of simultaneous linear equations $x+\lambda y-2 z=1, x-y+\lambda z=2$ and $x-2 y+3 z=3$ is inconsistent for $\lambda=\lambda_1$ and $\lambda_2$, then $\lambda_1+\lambda_2=$

The system of linear equation $(\sin \theta) x+y-2 z=0$, $2 x-y+(\cos \theta) z=0$ and $-3 x+(\sec \theta) y+3 z=0$, where $\theta \neq(2 n+1) \frac{\pi}{2}$, has non-trivial solution for

If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$, then $\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))$

The sum of all the roots of the equation

$\left|\begin{array}{ccc}x & -3 & 2 \\ -1 & -2 & (x-1) \\ 1 & (x-2) & 3\end{array}\right|=0$ is

If $\left|\begin{array}{ccc}1 & 2 & 3-\lambda \\ 0 & -1-\lambda & 2 \\ 1-\lambda & 1 & 3\end{array}\right|=A \lambda^3+B \lambda^2+C \lambda+D$, then $D+A=$

If $A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right]$ and $2 A-B=\left[\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right]$, then $\operatorname{tr}(A)-\operatorname{tr}(B)=$

$A, C$ are $3 \times 3$ matrices $B, D$ are $3 \times 1$ matrices. If $A X=B$ has unique solution and $C X=D$ has infinite number of solutions, then

$A$ and $B$ are two non-square matrices. If $P=A+B, Q=A^T B, R=A B^T$, then the matrices whose order is equal to the order of $A$ are

If the augmented matrix corresponding to the system of equations $x+y-z=1,2 x+4 y-z=0$ and $3 x+4 y+5 z=18$ is transformed to $\left[\begin{array}{cccc}1 & a & 0 & -1 \\ 0 & 2 & 1 & b \\ 0 & 0 & c & 32\end{array}\right]$ then $\sqrt{a+b+c}=$

If $\left|\begin{array}{ccc}9 & 25 & 16 \\ 16 & 36 & 25 \\ 25 & 49 & 36\end{array}\right|=K$, then $K, K+1$ are the roots of the equation

$A=\left[\begin{array}{ccc}1 & -3 & -5 \\ -2 & 4 & -6 \\ 7 & -11 & 13\end{array}\right]$, then $\sqrt{|\operatorname{adj} A|}=$

If $\Delta_r=\left|\begin{array}{cc}\frac{1}{3 r-2} & \frac{2}{3 r-5} \\ 0 & \frac{3}{3 r+1}\end{array}\right|$ then $\sum\limits_{r=1}^{33} \Delta_r=$

$A=\left[a_{i j}\right]$ is a $3 \times 3$ matrix with positive integers as its elements. Elements of $A$ are such that the sum of all elements of each row is equal to 6 and $a_{22}=2$.

If $\mathrm{a}_{i j}=\left\{\begin{array}{cl}\mathrm{a}_{i j}+\mathrm{a}_{j i}, & j=i+1 \text { when } i < 3 \\ \mathrm{a}_{i j}+\mathrm{a}_{j i}, & j=4-i \text { when } i=3\end{array}\right.$ for $i=1,2,3$, then $|\mathrm{A}|=$

If $A=\left[\begin{array}{lll}x & y & y \\ y & x & y \\ y & y & x\end{array}\right]$ is a matrix such that $5 A^{-1}=\left[\begin{array}{ccc}-3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3\end{array}\right]$, then $A^2-4 A=$

If $A=\left[\begin{array}{lll}9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2\end{array}\right]$ and $A A^T-A^2=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$, then $\sum\limits_{\substack{1 \leq i \leq 3 \\ 1 \leq j \leq 3}} a_{i j}=$

If $a \neq b \neq c, \Delta_1=\left[\begin{array}{lll}1 & a^2 & b c \\ 1 & b^2 & c a \\ 1 & c^2 & a b\end{array}\right]$, $\Delta_2=\left[\begin{array}{ccc}1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right]$ and $\frac{\Delta_1}{\Delta_2}=\frac{6}{11}$, then $11(a+b+c)=$

The system of equations $x+3 y+7=0$, $3 x+10 y-3 z+18=0$ and $3 y-9 z+2=0$ has

If $\mathrm{A}=\left[\begin{array}{lll}1 & 2 & 2 \\ 3 & 2 & 3 \\ 1 & 1 & 2\end{array}\right]$ and $\mathrm{A}^{-1}=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$, then $\sum_{\substack{1 \leq i \leq 3 \\ 1 \leq j \leq 3}} a_{i j}=$

The system of simultaneous linear equations

$$ \begin{aligned} & x-2 y+3 z=4,3 x+y-2 z=7 \\ & 2 x+3 y+z=6 \text { has } \end{aligned} $$

If $X_{4 \times 3}, Y_{4 \times 3}$ and $P_{2 \times 3}$ are the matrices, then the order of the matrix $\left[P\left(X^T Y\right)^{-1} P^T\right]^T$ is

If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]$ and $\alpha, \beta \in R$ are such that $\alpha A^2-\beta A=2 I$, then $\alpha^2+\beta=$

If $\left|\begin{array}{ccc}(1+\alpha)^2 & (1+2 \alpha)^2 & (1+3 \alpha)^2 \\ (2+\alpha)^2 & (2+2 \alpha)^2 & (2+3 \alpha)^2 \\ (3+\alpha)^2 & (3+2 \alpha)^2 & (3+3 \alpha)^2\end{array}\right|=k$ and $\alpha=-2$, then $k=$

7. If the system of equations $x+y+z=5, x+2 y+2 z=6$ and $x+3 y+\lambda z=\mu(\lambda, \mu \in R)$ is solvable by Matrix Inversion Method, then

If $A$ is a square matrix of order $3, \operatorname{then}\left|\operatorname{Adj}\left(\operatorname{Adj} A^2\right)\right|=$

If $A$ and $B$ are two square matrices of the same order and $(A B+B A)^T+(A B-B A)^T=2 B A$, then

If $\operatorname{adj}\left[\begin{array}{ccc}1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1\end{array}\right]=\left[\begin{array}{ccc}5 & m & -2 \\ 1 & 1 & 0 \\ -2 & -2 & n\end{array}\right]$, then $m+n=$

If $A=\left[\begin{array}{ll}0 & 3 \\ 0 & 0\end{array}\right]$ and $f(x)=x+x^2+x^3+\ldots \ldots+x^{2023}$, then $f(A)+I=$

4. If $A=\left[\begin{array}{lll}b & a & 0 \\ c & 0 & b \\ a & a & b\end{array}\right]$ and $B=\left[\begin{array}{lll}0 & a & b \\ b & 0 & c \\ b & a & a\end{array}\right]$ are two matrices such that $A B=\left[\begin{array}{ccc}2 & 2 & 7 \\ 1 & 8 & 5 \\ 3 & 6 & 10\end{array}\right]$, then $a^2+b^2+c^2=$

If $A=\left[\begin{array}{lll}1 & a & 3 \\ b & 2 & c \\ 3 & d & 4\end{array}\right]$ is a symmetric matrix and $B=\left[\begin{array}{ccc}0 & 5 & b \\ -5 & 0 & -7 \\ 6 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then $A B=$

If the inverse of the matrix $A=\left[\begin{array}{ccc}-1 & -3 & -2 \\ 0 & 1 & 2 \\ 3 & 4 & 5\end{array}\right]$ is $A^{-1}=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$, then $a_1+c_2+b_3=$

If $x=\alpha, y=\beta, z=\gamma$ is the unique solution of the system of linear equations $2 x-3 y+5 z=12,5 x+2 y+3 z=11$ and $x+2 y-3 z=-3$, then $2 \alpha+5 \beta+3 \gamma=$

If $A=\left[\begin{array}{ccc}1 & 2 & -1 \\ -1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}-3 & -2 & 4 \\ 2 & 2 & -1 \\ -2 & 0 & 3\end{array}\right]$, then $A^2=$

$$ \left|\begin{array}{lll} 2 & 3 & 5 \\ 3 & 5 & 2 \\ 5 & 2 & 3 \end{array}\right|+\left|\begin{array}{ccc} 1 & 1 & 1 \\ 7 & 11 & 13 \\ 49 & 121 & 169 \end{array}\right|= $$

If $A=\left[\begin{array}{ccc}k & 5 & 2 \\ 2 & -k & 5 \\ 5 & 2 & -k\end{array}\right]$ and $\operatorname{det} A=190$, then $\operatorname{adj} A=$

If the unique solution of the simultaneous linear equations $3 x-2 y+z=5 k, 2 x+3 y-2 z=-5 k$, $x+4 y+3 z=k$ is $x=\alpha, y=\beta, z=3$, then $k=$

$$ \left|\begin{array}{ccc} \sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5 \end{array}\right|= $$

If $A$ is a non-singular matrix such that $(A-2 I)$ $(A-3 I)=0$, then $\frac{1}{5} A+\frac{6}{5} A^{-1}=$

Let $A$ be a matrix such that $A B$ is a scalar matrix, where $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ and $\operatorname{det}(3 A)=27$. Then, $3 A^{-1}+A^2=$

If $A$ is a symmetric matrix with real entries, then

$$ \begin{aligned} &\text { If } \omega \neq 1 \text { is a cube root of unity, then }\\ &\left|\begin{array}{ccc} \omega+\omega^2 & \omega^2+\omega^9 & \omega^9+\omega \\ \omega^{27}+\omega^{31} & \omega^{31}+\omega^{17} & \omega^{17}+\omega^{27} \\ \omega^{30}+\omega^{41} & \omega^{41}+\omega^{19} & \omega^{19}+\omega^{30} \end{array}\right|= \end{aligned} $$

If the system of equations

$x+k y+3 z=-2$,

$4 x+3 y+k z=14,$

If $A$ is a $2 \times 2$ matrix such that $\operatorname{det} A=-21$ and trace of $A^3$ is 2024 , then the trace of $A$ is

If $\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$ is a skew-symmetric matrix and $b, c$ and $f$ are non-zero real numbers, then $\frac{b}{c}=$

In the matrix $\left[\begin{array}{ccc}-1 & x & 3 \\ -4 & -5 & -6 \\ -7 & y & 9\end{array}\right]$, if the cofactors of -6 and -7 are respectively 22 and 27 , then $5 x+y=$

Consider the simultaneous linear equations $\beta x+\alpha y-z=-1,3 x-\beta y+\alpha z=0 \alpha x+\beta y+z=1$, In the usual notation used in Crammer's rule, given that $\frac{\Delta_1}{\Delta}=-1, \frac{\Delta_2}{\Delta}=1, \frac{\Delta_3}{\Delta}=2$, then $(\alpha, \beta)=$

If $\left|\begin{array}{cc}2+3 i & i \\ 1-2 i & -i\end{array}\right|=x+i y$, then $x+y=$

$A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 3 & 2\end{array}\right]$, then $\left(A+A^T\right)\left(A-A^T\right)=$

If $f(x)=\left|\begin{array}{ccc}x & x+1 & x+3 \\ x+2 & x+4 & x+7 \\ x+6 & x+9 & x+13\end{array}\right|$, then $f(5)=$

Let $A=\left[\begin{array}{lll}2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2\end{array}\right]$. If $A^{-1}=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$ and $\gamma$ are real numbers and $I$ is a $3 \times 3$ identity matrix, then $17 \alpha+5 \beta+\gamma=$

For a system of simultaneous linear equations, if $A X=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right], \operatorname{Adj} A=\left[\begin{array}{ccc}1 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & 1\end{array}\right]$ and $\operatorname{det} A>0$, then $X=$

Let $A=\left[\begin{array}{ll}0 & 1 \\ 1 & k\end{array}\right], k \in R$ and $A^3=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$. If $d=228$, then $b+c=$

Let $A$ and $B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ unit matrix such that $A B-C$ is a non-singular matrix. Let $D=(A B-C)^{-1}$. Then, consider the following statements.

Statement I $\operatorname{det}(B A)=\operatorname{det}(B A-C) \operatorname{det}(B D A)$

Statement II $A B D=D A B$

Which of the above statements is (are) true?

Let $A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right], B=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$, then $\left(A^{-1} B\right)^{-1}+\left(A B^{-1}\right)^{-1}=$

Let $\alpha, \beta$ and $\gamma$ be real numbers.

If $\left[\begin{array}{ccc}7 & 5 & \alpha \\ \beta & 2 & 11 \\ 3 & \gamma & 1\end{array}\right]\left[\begin{array}{l}1 \\ 3 \\ 2\end{array}\right]=\left[\begin{array}{c}\alpha+\beta \\ -2 \alpha+\beta-2 \gamma \\ \alpha+2 \beta+3 \gamma\end{array}\right]$, then $100+\frac{2 \alpha+11 \beta}{\gamma}=$

If $\left[\begin{array}{ccc}0 & 2 & a \\ b & 0 & 4 \\ -3 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then $\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\left[\begin{array}{ll}b & c \\ c & b\end{array}\right]=$

If $\left[\begin{array}{ccc}-1 & 2 & b \\ a & 5 & 6 \\ 3 & c & 7\end{array}\right]$ is a symmetric matrix, then $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=$

If the matrix $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$ satisfies the matrix equation $A^2-4 A-5 I=0$, then $A^{-1}=$

Consider the simultaneous linear equations $A X=B$ and $A Y=Q$. If $A$ is an invertible matrix and $B$ is the unique solution of $A Y=Q$, then the solution of $A X=B$ is

If $f(x)=\left|\begin{array}{ccc}-\sin x & 2 \sin 2 x & 4 \cos ^2 x \\ \cos x & 4 \sin ^2 x & 2 \sin 2 x \\ 0 & -\cos x & \sin x\end{array}\right|$, then $f\left(\frac{5 \pi}{4}\right)+f^{\prime}\left(\frac{5 \pi}{4}\right)=$

If $A+B=\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 2 & 0 \\ 0 & 2 & 2\end{array}\right], A B=\left[\begin{array}{lll}1 & 2 & 2 \\ 1 & 1 & 0 \\ 1 & 2 & 1\end{array}\right]$, then $A^2+B(A+B)=$

If $A, P, B$ are $3 \times 3$ matrices. If $|-B|=5,\left|B A^T\right|=15$, $\left|P^T A P\right|=-27$, then one of the values of $|P|$ is

If $A$ is a $3 \times 3$ matrix and $|A|=\frac{1}{2}$, then $\left|A^{-1}(\operatorname{Adj}(\operatorname{Adj} A))\right|^{-1}=$

Let $x=\alpha, y=\beta, z=\gamma$ be the unique solution of the system of simultaneous linear equations $2 x+3 y-2 z+4=0,3 x-4 y+3 z+5=0$, $k x-2 y+z+3=0$. If $\alpha=-2$, then $k=$

20. If $\frac{x^2+7}{\left(x^2+1\right)(x-2)}=\frac{A}{x-2}+\frac{B x+C}{x^2+1}$, then the determinant of the matrix $\left[\begin{array}{ll}A & B \\ C & \frac{2}{5}\end{array}\right]$ is

$\left|\begin{array}{ccc}1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right|=$

Let $\alpha, \beta, \gamma$ be real numbers. If $A=\left[\begin{array}{ccc}7 & 3 & \alpha \\ \beta & 1 & -11 \\ -5 & \gamma & 19\end{array}\right]$ is a $3 \times 3$ matrix satisfying $A\left[\begin{array}{c}5 \\ -13 \\ 11\end{array}\right]=\left[\begin{array}{c}-290 \\ -119 \\ 210\end{array}\right]$, then $(\operatorname{adj} A)^{-1}+\operatorname{adj} A^{-1}=$

If $[\alpha \beta \gamma]\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & -5\end{array}\right]=[352]$, then $\alpha^3+\beta^3+\gamma^3=$

If $a$ and $b$ are any two real numbers, then

$$ \left|\begin{array}{ccc} 2 a-2 b-4 & 4 a & 4 a \\ 4 & 2-b-a & 4 \\ 2 b & 2 b & b-a-2 \end{array}\right|= $$

Let $A=\left[\begin{array}{ccc}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x\end{array}\right]$ and $A^2=A$. If $r$ is the rank of $A$, then $r+x=$

Let $a, b, c, d \in \mathbf{R}$ be such that $a d-b c \neq 0$ and $e$ be a positive number other than 1 .

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|, \Delta_2=\left|\begin{array}{cc}a & m \\ c & n\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$, then the values of $x$ and $y$ are respectively.

For a square matrix $B$ of order 3 , if $B^T=B^{-1}$ and $|B|=1$, then $|B-I|=$

For $\alpha, \beta \in[0,2 \pi]$ and $\gamma \in[0, \pi)$ consider the system of equations

$$ \begin{aligned} & 2 \sin \alpha-\cos \beta+3 \tan \gamma=3 \\ & 4 \sin \alpha+2 \cos \beta-2 \tan \gamma=2 \\ & 6 \sin \alpha-3 \cos \beta+\tan \gamma=9 \end{aligned} $$

Then, which one of the following is true?

$$ \text { The rank of } A=\left[\begin{array}{ccc} 1 & x & x+1 \\ 2 x & x^2-x & x^2+x \\ 3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right) \end{array}\right] \text { is } $$

Let $I$ be a unit matrix of order 6 . Let $A=\left(a_{i j}\right)$ be a square matrix of order 6 such that $a_{i j}=\left\{\begin{array}{l}1, \text { if } i+j=7 \\ 0, \text { if } i+j \neq 7\end{array}\right.$ then $\left(A(\operatorname{adj} A) A^{-1}\right) A^2=$

Let $a, b, c \notin\{0,1\}$. If the system of equations

$$ \begin{aligned} & \Pi_1 \equiv x+a y+a z=0 \\ & \Pi_2 \equiv b x+y+b z=0 \\ & \Pi_3 \equiv c x+c y+z=0 \end{aligned} $$

has a non-trivial solution, then the system of equations $\Pi_1=a, \Pi_2=b, \Pi_3=c$ has

$A$ is a singular matrix of order five. $B$ is another matrix having the rank $\rho(B)$ equal to the $\operatorname{rank} \rho(A)$ and $B$ has a non-zero minor of order 3. Then which one of the following is true?

A value of $\theta$ in $\left(0, \frac{\pi}{2}\right)$ and satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \sin 4 \theta\end{array}\right|=0$ is

Let $[A]_{3 \times 3}$ be a non-singular matrix such that

$$ A^{-1}=\frac{1}{3}\left(A^2-5 A+7 I\right) . $$

Then $17 A^8-85 A^7+119 A^6-51 A^5-19 A^4+95 A^3-133 A^2+58 A+I=$

If $\left[\begin{array}{ccc}2 & 1 & 1 \\ 0 & 3 & -1 \\ 1 & -1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]$, then $\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=$

Let $A=\left[\begin{array}{ccc}1 & 4 & 2 \\ 2 & -1 & 4 \\ -3 & 7 & -6\end{array}\right]$ and $B=\left[b_{i j}\right]_{3 \times 3}$ with $b_{11}=2$, $b_{13}=-2, b_{12}=0$ is such that $A B=\left[\begin{array}{ccc}2 & 14 & -4 \\ 4 & 1 & -8 \\ -6 & 15 & 12\end{array}\right]$, then $|B|+\operatorname{trace}(B)=$

A is a $m \times n$ matrix of rank 4 . If A contains an $m$ th order non singular sub matrix and $A^T A$ is a $7 \times 7$ matrix, then the number of rows of $A$ is

If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real number $\mathbf{R}$ such that $C D=-D C$, then $n$ is

If $A, B$ are two non singular matrices of order $3,|B|=k$, a positive integer, then match the items of list-I with the items of list-II.

$$ \text { The correct match is } $$

All the real values of $p, q$ so that the system of equations

$$ 2 x+p y+6 z=8, x+2 y+q z=5 $$

and $\quad x+y+3 z=4$

may have no solution are

If $p$ and $q$ are two distinct real values of $\lambda$ for which the system of equations

$$ \begin{array}{r} (\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0 \\ (\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0 \\ 2 x+(3 \lambda+1) y+3(\lambda-1) z=0 \end{array} $$

has non-zero solution, then $p^2+q^2-p q=$