Matrices practice · Mixed difficulty

Assertion : If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$, then $B$ is the inverse of ${A}$.

Reason: If $A$ is a square matrix of order $m$ and if there exists another square matrix ${B}$ of the same order ${m}$, such that ${AB}={BA}={I}$, then ${B}$ is called the inverse of ${A}$.

If $\mathrm{A}$ is an invertible matrix of order 2 , then $\operatorname{det} .\left(\mathrm{A}^{-1}\right)$ is equal to :

The value of $\Delta=\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \sin (\alpha+\delta) \\ \sin \beta & \cos \beta & \sin (\beta+\delta) \\ \sin \gamma & \cos \gamma & \sin (\gamma+\delta)\end{array}\right|,$ is

Let $A$ be a $2 \times 2$ matrix with $\operatorname{det}(A)=-1$ and $\operatorname{det}$ $((A+I)(\operatorname{Adj}(A)+I))=4$. Then the sum of the diagonal elements of $A$ can be

Let $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ and $C\left(x_3, y_3\right)$ be vertices of an equilateral triangle whose side is 4 units. Let $\Delta=\left|\begin{array}{lll} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array}\right|,$ then $\Delta^2$ is equal to

The matrix $A$ satisfying $A\left[\begin{array}{ll}1 & 5 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -1 \\ 6 & 0\end{array}\right]$ is

Let $A=\left[a_{i j}\right]$ be a square matrix of order 3 such that $a_{i j}=2^{j-i}$, for all $i, j=1,2,3$. Then, the matrix $A^2+A^3$ $+\ldots+A^{10}$ is equal to :

If $a,b,c,$ are in A.P., then the value of $\left| {\begin{array}{cccccccccccccccccccc} {x + 1}&{x + 2}&{x + a}\\ {x + 2}&{x + 3}&{x + b}\\ {x + 3}&{x + 4}&{x + c} \end{array}} \right|,$ is

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 solution, if $k \ne 2,\,\,\,\,k \ne - 2$
B) The system has unique solution, if $k = - 2$.
C) The system has unique solution, if $k = 2$.
D) The system has no solution, if $k = 2$.
E) The system has infinite number of solutions, if $k \ne - 2.$
Which of the following statements are correct?

If $\left[ {\begin{array}{cccccccccccccccccccc} 1&1\\ 0&1 \end{array}} \right] \cdot \left[ {\begin{array}{cccccccccccccccccccc} 1&2\\ 0&1 \end{array}} \right] \cdot \left[ {\begin{array}{cccccccccccccccccccc} 1&3\\ 0&1 \end{array}} \right] \cdot \cdot \cdot \cdot \left[ {\begin{array}{cccccccccccccccccccc} 1&{n - 1}\\ 0&1 \end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc} 1&{78}\\ 0&1 \end{array}} \right],$ then the inverse of $\left[ {\begin{array}{cccccccccccccccccccc} 1&n\\ 0&1 \end{array}} \right]$ is

If $a, b$ and $c$ are sides of $\triangle A B C$ such that
$\left|\begin{array}{ccc}c & b \cos B+c \beta & a \cos A+b \alpha+c \gamma \\ a & c \cos B+a \beta & b \cos A+c \alpha+a \gamma \\ b & a \cos B+b \beta & c \cos A+a \alpha+b \gamma\end{array}\right|=0$ (where $\alpha, \beta, \gamma \in R^{+}$and $\angle A, \angle B, \angle C \neq \frac{\pi}{2}$ ), $\triangle A B C$ is

The value of $\left|\begin{array}{ccc}a^2 & -a b & -a c \\ -a b & b^2 & -b c \\ c a & b c & -c^2\end{array}\right|,$ is

If two matrices $\mathrm{A}$ and $\mathrm{B}$ are of order $p \times q$ and $r \times s$ respectively, can be subtracted only, if

The system of equation
$\begin{array}{l} \alpha x + y + z = \alpha - 1\\ x + \alpha y + z = \alpha - 1\\ x + y + \alpha z = \alpha - 1 \end{array}$
Has no solution, if $\alpha$ is

Given that $x y z=-1,$ the value of the determinant $\left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|,$ is

Which one of the following is correct?

The value of $\left| {\begin{array}{cccccccccccccccccccc} {b + c}&a&a\\ b&{c + a}&b\\ c&c&{a + b} \end{array}} \right|,$ is

If $\log _3 x(x+2)=1$, then $x$ is equal to

If ${I_n}$ is the identity matrix of order $n$, then ${\left( {{I_n}} \right)^{ - 1}}$ is equal to

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

Consider the following statements

I. If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

II. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant remains same.

If ${a_1},{a_2}_. \ldots ..,{a_n}, \ldots ..,$ are in GP, then the determinant $\Delta = \left| {\begin{array}{cccccccccccccccccccc} {\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\ {\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\ {\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}} \end{array}} \right|$ is equal to

If $\mathrm{A}$ is a square matrix $\mathrm{A}+\mathrm{A}^{\mathrm{T}}$ is symmetric matrix, then $\mathrm{A}-\mathrm{A}^{\mathrm{T}}=$

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x=c y+b z, y=a z+c x$ and $z=b x+a y$. Then, $a^2+b^2+c^2+2 a b c$ is equal to

If adj $\mathrm{B}=\mathrm{A},|\mathrm{P}|=|\mathrm{Q}|=1$, then adj $\left(\mathrm{Q}^{-1} \mathrm{BP}^{-1}\right)$ is

If $A$ is a $3 \times 3$ non-singular matrix such that $A{A^T} = {A^T}A$ and $B = {A^{ - 1}}{A^T},$ then $B{B^T}$ is equal to

If$I$ is unit matrix of order 10, then the determinant of $I$ is equal to

If $\Delta_{1}=\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ and $\Delta_{2}=\left|\begin{array}{ccc}x & \sin 2 \theta & \cos 2 \theta \\ -\sin 2 \theta & -x & 1 \\ \cos 2 \theta & 1 & x\end{array}\right|, x \neq 0$; then for all $\theta \in\left(0, \frac{\pi}{2}\right)$

Consider the following relation $\mathrm{R}$ on the set of real square matrices of order 3 .
$R=\left\{(A, B) \mid A=P^{-1} B P\right.$ for some invertible matrix $\left.P\right\}$
Statement-1: $R$ is equivalence relation.

Statement-2 : For any two invertible $3 \times 3$ matrices $M$ and
$N,(M N)^{-1}=N^{-1} M^{-1} \text {. }$

If one root of determinant $\left| {\begin{array}{cccccccccccccccccccc} x&3&7\\ 2&x&2\\ 7&6&x \end{array}} \right| = 0$, is $- 9,$ then the other two roots are