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 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. $

Then which of the following statements is (are) TRUE?

Then which of the following statements is (are) TRUE?

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

$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $

is

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 statements is (are) TRUE?

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

$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$

The correct option is:

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 $P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$, where $\alpha$ $\in$ R. Suppose $Q = [{q_{ij}}]$ is a matrix such that PQ = kl, where k $\in$ R, k $\ne$ 0 and I is the identity matrix of order 3. If ${q_{23}} = - {k \over 8}$ and $\det (Q) = {{{k^2}} \over 2}$, then

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 ${{{q_{31}} + {q_{32}}} \over {{q_{21}}}}$ equals

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?

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}} \cr {{{(3 + \alpha )}^2}} & {{{(3 + 2\alpha )}^2}} & {{{(3 + 3\alpha )}^2}} \cr } } \right| = - 648\alpha $ ?

Let $\omega$ be a complex cube root of unity with $\omega$ $\ne$ 1 and P = [p_ij] be a n $\times$ n matrix with p_ij = $\omega$^{i + j}. Then P² $\ne$ 0, when n = ?

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 P^T = 2P + I, where P^T is the transpose of P and I is the 3 $\times$ 3 identity matrix, then there exists a column matrix $X = \left[ {\matrix{ x \cr y \cr z \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$ such that

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 and N be two 3 $\times$ 3 non-singular skew symmetric matrices such that MN = NM. If P^T denotes the transpose of P, then M²N²(M^TN)^$-$1(MN^$-$1)^T is equal to

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 value of ${3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}$ is equal to

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 $\omega$ $\ne$ 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, b, and c is either $\omega$ or $\omega$². Then the number of distinct matrices in the set S is

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is

[Note : The trace of a matrix is the sum of its diagonal entries.]

The number of matrices in A is

The number of matrices A in A for which the system of linear equations $A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$ has a unique solution, is

The number of matrices A in A for which the system of linear equations $A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$ is inconsistent, is

Consider the system of equations:

$x-2y+3z=-1$

$-x+y-2z=k$

$x-3y+4z=1$

Statement - 1 : The system of equations has no solution for $k\ne3$.

and

Statement - 2 : The determinant $\left| {\matrix{ 1 & 3 & { - 1} \cr { - 1} & { - 2} & k \cr 1 & 4 & 1 \cr } } \right| \ne 0$, for $k \ne 3$.

The sum of the elements of $\mathrm{U}^{-1}$ is:

The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right] U\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is :