Matrices practice · Easy

If $\left[\begin{array}{cc}i & 0 \\ 3 & -i\end{array}\right]+X=\left[\begin{array}{cc}i & 2 \\ 3 & 4+i\end{array}\right]-X$, then $X$ 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 $\theta \in\left(0, \frac{\pi}{2}\right)$

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations

$x-y+z=5$

$2 x+2 y+\alpha z=8$

$3 x-y+4 z=\beta$

Has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

If the trivial solution is the only solution of the system of equations
$x-k y+z=0$

$k x+3 y-k z=0$
$3 x+y-z=0$, then the set of values of $k$, is

$\left| {\begin{array}{cccccccccccccccccccc} 1&5&\mathrm{\pi }\\ {{{\log }_e}e}&5&{\sqrt 5 }\\ {{{\log }_{10}}10}&5&e \end{array}} \right|$ is equal to

Let $a, b, c$ be such that $b(a+c) \neq 0.$ If $\left|\begin{array}{rrr}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|+\left|\begin{array}{ccc}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^n c\end{array}\right|=0$ then the value of $n,$ is

If $m\left[ { - 3\;\;\;4} \right] + n\left[ {4\;\; - 3} \right] = \left[ {10\;\;\; - 11} \right],\;\mathrm{then}\;\,3m + 7n\;$ is equal to

For which of the following ordered pairs $\left( {\mu ,\,\delta } \right),$ the system of liner equations $x + 2y + 3z = 1,\,\,\,\,3x + 4y + 5z = \mu ,\,\,\,4x + 4y + 4z = \delta$ is inconsistent?

The matrix $P=\left[\begin{array}{ccc}0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0\end{array}\right]$ is

If $A$ is a non-singular matrix of order $3 \times 3$, then $\operatorname{adj}(\operatorname{adj} A)$ is equal to

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

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 ${A}=\left[\begin{array}{cc}4 & 4 \\ -3 & 4\end{array}\right] ; {B}=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, then $\left({B}^{-1} {A}^{-1}\right)^{-1}=$

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

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?

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

Let $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha\end{array}\right]$ and $|2 A|^3=2^{21}$ where $\alpha, \beta \in {Z}$, then a value of $\alpha$ is

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.