Matrices and Determinants

Consider the matrix $$ M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}. $$ Let $p, q, r, s, a, b, c$ and $d$ be integers such that $$ M^{26} = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \quad \text{and} \quad \sum\limits_{k=1}^{26} M^k ...

Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

Let $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ and $P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$. Let $Q=\left(\begin{array}{ll}x & y \\ z & 4\end{array}\right)$ for some non-zero real numbers $x, y$, and $z$,...

Consider the matrix$$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that$$ ...

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M=\left(a_{i j}\right)_{3 \times 3}$, define $R_i=a_{i 1}+a_{i 2}+a_{i 3}$ and $C_j=a_{1 j}+a_...

Let $S=\left\{A=\left(\begin{array}{lll}0 & 1 & c \\ 1 & a & d \\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.$ and $\left.|A| \in\{-1,1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in ...

Let $\mathbb{R}^2$ denote $\mathbb{R} \times \mathbb{R}$. Let

\(S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .\)

Then which of the following st...

Let $R=\left\{\left(\begin{array}{lll}a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0\end{array}\right): a, b, c, d \in\{0,3,5,7,11,13,17,19\}\right\}$.
Then the number of invertible matrices in $R$ is :

Let $M=\left(a_{i j}\right), i, j \in\{1,2,3\}$, be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$, otherwise $a_{i j}=0$. Then which of the following statements is(are) true?

Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$$ \begin{aligned} & x+2 y+z=7 \\\\ & x+\alpha z=11 \\\\ & 2 x-3 y+\beta z=\gamma \end{aligned} $$
Match each entry in List-I to the correct...

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 equal to $M^{2022} ?$

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 $A^{7}-(\beta-1) A^{6}-\beta A^{5}$ is a singular matrix, then the value of $9 \beta$ is __...

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 harmonic progression. Consider the system of linear equations
$$$
\begin{gathered}
x+y+z=1 \\...

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 matrices such that (I \(-\) EF) is invertible. If G = (I \(-\) EF)\(-\)1, then which of the...

For any 3 \(\times\) 3 matrix M, let | M | denote the determinant of M. Let\(E = \left[ {\matrix{ 1 & 2 & 3 \cr 2 & 3 & 4 \cr 8 & {13} & {18} \cr } } \right]\), $$P = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 0 & 1 \cr
...

Let \(\alpha\), \(\beta\) and \(\gamma\) be real numbers such that the system of linear equationsx + 2y + 3z = \(\alpha\)4x + 5y + 6z = \(\beta\)7x + 8y + 9z = \(\gamma\) \(-\) 1is consistent. Let | M | represent the determinant of the mat...

Let \(\alpha\), \(\beta\) and \(\gamma\) be real numbers such that the system of linear equationsx + 2y + 3z = \(\alpha\)4x + 5y + 6z = \(\beta\)7x + 8y + 9z = \(\gamma\) \(-\) 1is consistent. Let | M | represent the determinant of the mat...

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 trace of A3 is \(-\)18, then the value of the determinant of A is .............

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 which of the following statements is/are ALWAYS TRUE?

$${P_1} = I = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 \cr

} } \right],\,{P_2} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 1 & 0 \cr

} } \right],\,{P_3} = \left[ {\matrix{
0 & 1 & 0 ...

Suppose
det\(\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k}.k} } & {\sum\limits_{k = 0}^n {{}^n{C_k}{3^k}} } \cr } } \right| = 0\)
holds for some p...

Let x \(\in\) R and let \(P = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]\), \(Q = \left[ {\matrix{ 2 & x & x \cr 0 & 4 & 0 \cr x & x & 6 \cr } } \right]\) and R = PQP\(-\)1, w...

Let \(M = \left[ {\matrix{ 0 & 1 & a \cr 1 & 2 & 3 \cr 3 & b & 1 \cr } } \right]\) andadj \(M = \left[ {\matrix{ { - 1} & 1 & { - 1} \cr 8 & { - 6} & 2 \cr { - 5} & 3 & { - 1} \cr } } \right]\)where a and b...

Let \(M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \alpha I + \beta {M^{ - 1}}\),where \(\alpha\) = $$\...

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 determinant of P is ............ .

Let S be the set of all column matrices \(\left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr {{b_3}} \cr } } \right]\) such that \({b_1},{b_2},{b_3} \in R\) and the system of equations (in real variables)$$\eqalign{
& - x + 2y + 5...

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?

For a real number \(\alpha\), if the system$$\left[ {\matrix{
1 & \alpha & {{\alpha ^2}} \cr
\alpha & 1 & \alpha \cr
{{\alpha ^2}} & \alpha & 1 \cr

} } \right]\left[ {\matrix{
x \cr
y \cr
z \cr

} } \ri...

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 is(are) correct?

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 \(Q = [{q_{ij}}]\) is a matrix such that \({P^{50}} - Q = I\) and $${{{q_{31}} + {q_{32}}} \o...

Let \(z = {{ - 1 + \sqrt 3 i} \over 2}\), where \(i = \sqrt { - 1}\), and r, s \(\in\) {1, 2, 3}. Let \(P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]\) and I be the identity matrix of...

The total number of distinct x \(\in\) R for which
\(\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10\) is ______________.

Let \(P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]\), where \(\alpha\) \(\in\) R. Suppose \(Q = [{q_{ij}}]\) is a matrix such that PQ = kl, where k \(\in\) R, k \(\ne\) 0 a...

Which of the following values of \(\alpha\) satisfy the equation
$$\left| {\matrix{
{{{(1 - \alpha )}^2}} & {{{(1 + 2\alpha )}^2}} & {{{(1 + 3\alpha )}^2}} \cr
{{{(2 + \alpha )}^2}} & {{{(2 + 2\alpha )}^2}} & {{{(2 + 3\alpha )}^2}} ...

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 matrices is(are) skew symmetric?

Let M and N be two 3 \(\times\) 3 matrices such that MN = NM. Further, if M \(\ne\) N2 and M2 = N4, then

Let M be a 2 \(\times\) 2 symmetric matrix with integer entries. Then, M is invertible, if

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 \(\ne\) 0, when n = ?

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 possible value(s) of the determinant of P is(are)

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 column matrix $$X = \left[ {\matrix{
x \cr
y \cr
z \cr

} } \right] \ne \...

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 determinant of P is 2, then the determinant of the matrix Q is

Let M be a 3 \(\times\) 3 matrix satisfying \(M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right]\), $$M\left[ {\matrix{
1 \cr
{ - 1} \cr
0 \c...

Let \(\omega\) \(\ne\) 1 be a cube root of unity and S be the set of all non-singular matrices of the form \(\left[ {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right]\), where each of a,...

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 \(\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}}\) 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 value of \({3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}\) is ...

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 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\(-\)1)T is equal to

Let $k$ be a positive real number and let
$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{a...