Mathematics
Matrices and Determinants
Previous Years Questions

## Numerical

Let $\beta$ be a real number. Consider the matrix $$A=\left(\begin{array}{ccc} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{array}\right)$$ If \$...
The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 $$\times$$ 2 matrix such that the trace of A is 3 and the...
Suppose det$$\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k... Let P be a matrix of order 3$$ \times $$3 such that all the entries in P are from the set {$$-$$1, 0, 1}. Then, the maximum possible value of the de... For a real number$$\alpha $$, if the system$$\left[ {\matrix{ 1 & \alpha & {{\alpha ^2}} \cr \alpha & 1 & \alpha \cr ...
The total number of distinct x $$\in$$ R for which $$\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3... Let$$z = {{ - 1 + \sqrt 3 i} \over 2}$$, where$$i = \sqrt { - 1} $$, and r, s$$\in$${1, 2, 3}. Let$$P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^...
Let M be a 3 $$\times$$ 3 matrix satisfying $$M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2... ## MCQ (Single Correct Answer) If M=\left(\begin{array}{rr}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{array}\right), then which of the following matrices is equa... Let$$p, q, r$$be nonzero real numbers that are, respectively, the$$10^{\text {th }}, 100^{\text {th }}$$and$$1000^{\text {th }}$$terms of a har... Let$$M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\co...
How many 3 $$\times$$ 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of MTM is 5?
Let $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$ and I be the identity matrix of order 3. If $$Q... If P is a 3$$\times$$3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3$$\times$$3 identity matrix, then there exists a... Let$$P = [{a_{ij}}]$$be a 3$$\times$$3 matrix and let$$Q = [{b_{ij}}]$$, where$${b_{ij}} = {2^{i + j}}{a_{ij}}$$for$$1 \le i,j \le 3$$. If the... If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is Let$$\omega$$be a solution of$${x^3} - 1 = 0$$with$${\mathop{\rm Im}\nolimits} (\omega ) > 0$$. If a = 2 with b and c satisfying (E), then the va... Let b = 6, with a and c satisfying (E). If$$\alpha$$and$$\beta$$are the roots of the quadratic equation ax2 + bx + c = 0, then$$\sum\limits_{n = ...
Let $$\omega$$ $$\ne$$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $$\left[ {\matrix{ 1 & a & b \cr \o... ## MCQ (More than One Correct Answer) For any 3$$\times$$3 matrix M, let | M | denote the determinant of M. Let$$E = \left[ {\matrix{ 1 & 2 & 3 \cr 2 & 3 & 4 \cr...
For any 3 $$\times$$ 3 matrix M, let |M| denote the determinant of M. Let I be the 3 $$\times$$ 3 identity matrix. Let E and F be two 3 $$\times$$ 3 m...
Let M be a 3 $$\times$$ 3 invertible matrix with real entries and let I denote the 3 $$\times$$ 3 identity matrix. If M$$-$$1 = adj(adj M), then w...
Let x $$\in$$ R and let $$P = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$, $$Q...$${P_1} = I = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right],\,{P_2} = \left[ {\matri...
Let $$M = \left[ {\matrix{ 0 & 1 & a \cr 1 & 2 & 3 \cr 3 & b & 1 \cr } } \right]$$ andadj $$M = \left[ {\matr... Let S be the set of all column matrices$$\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]$$such that$${b_1},{b_2},...
Which of the following is(are) NOT the square of a 3 $$\times$$ 3 matrix with real entries?
Let a, $$\lambda$$, m $$\in$$ R. Consider the system of linear equations ax + 2y = $$\lambda$$ 3x $$-$$ 2y = $$\mu$$ Which of the following statements...
Let $$P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha$$ $$\in$$ R. ...
Let X and Y be two arbitrary, 3 $$\times$$ 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 $$\times$$ 3, non-zero, symmetric matrix. Then...
Which of the following values of $$\alpha$$ satisfy the equation $$\left| {\matrix{ {{{(1 - \alpha )}^2}} & {{{(1 + 2\alpha )}^2}} & {{{(1 + 3\alph... Let M be a 2$$\times$$2 symmetric matrix with integer entries. Then, M is invertible, if Let M and N be two 3$$\times$$3 matrices such that MN = NM. Further, if M$$\ne$$N2 and M2 = N4, then Let$$\omega$$be a complex cube root of unity with$$\omega\ne$$1 and P = [pij] be a n$$\times$$n matrix with pij =$$\omega$$i + j. Then P2 ... For 3 × 3 matrices M and N, which of the following statement(s) is(are) NOT correct? If the ad joint of a 3$$\times$$3 matrix P is$$\left[ {\matrix{ 1 & 4 & 4 \cr 2 & 1 & 7 \cr 1 & 1 & 3 \cr } } \right]$$, then the ... Let M and N be two 3$$\times$$3 non-singular skew symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M2N2(MTN)$$-$$1(MN$$-...
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