Matrices practice · Easy

If every element of a third order determinant of value $\Delta$ is multiplied by 5, then the value of new determinant, is

If $x, y, z$ are real and
$A=\left[\begin{array}{ccc} 0 & 2 y & z \\ x & y & -z \\ x & -y & z \end{array}\right]$
is such that $A A^{\mathrm{T}}=I$, then

Assuming that the sum and product given below are defined, which of the following is not true for matrices?

Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right], a, b, c, d \in {R}.$ If $A^5=A^3+I$, then $A$ is

Let $A=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]$ satisfies $A^2+a A+b I=O$, then $a, b$ are respectively equal to

The system of linear equations $x + y + z = 2,2x + 3y + 2z = 5$
$2x + 3y + ({a^2} - 1)z = a + 1$

The number of solutions of the matrix equation $X^2=I$ other than $I$ is

Let $A=\left[\begin{array}{cc}2 & -1 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{ll}5 & 2 \\ 7 & 4\end{array}\right], C=\left[\begin{array}{ll}2 & 5 \\ 3 & 8\end{array}\right]$. Let $D$ be a matrix such that $C D=A B$, then $D$ equals

Let
$A=\left[\begin{array}{rr} 2 & 3 \\ -1 & 4 \\ 1 & 0 \end{array}\right], B=\left[\begin{array}{rr} -1 & 2 \\ 3 & 1 \\ 5 & 4 \end{array}\right] \text { and } C=\left[\begin{array}{rr} 2 & -1 \\ 2 & 3 \\ -4 & 1 \end{array}\right]$

If
$A-B+x C=\left[\begin{array}{cc} 7 & -1 \\ 0 & 9 \\ -12 & -2 \end{array}\right]$
then the value of $x$ is

If $a_1\, b_1\, c_1, a_2\, b_2\, c_2$ and $a_3\, b_3\, c_3$ are three digit even natural numbers and
$\Delta=\left|\begin{array}{lll}c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3\end{array}\right|$, then $\Delta$ is

Let $P = \left[ {\begin{array}{cccccccccccccccccccc} 1&0&0\\ 3&1&0\\ 9&3&1 \end{array}} \right]\mathrm{ }\,\,\mathrm{and}\,\,\mathrm{ }Q = \left[ {{q_{ij}}} \right]\mathrm{ }\,\,\mathrm{be}\,\,\mathrm{ two }\,\,3 \times 3$ matrices such that $Q - {P^5} = {I_3}.$ Then, $\frac{{{q_{21}} + {q_{31}}}}{{{q_{32}}}}$ is equal to

If $\Delta=\left|\begin{array}{ccc}\cos \frac{\theta}{2} & 1 & 1 \\ 1 & \cos \frac{\theta}{2} & -\cos \frac{\theta}{2} \\ -\cos \frac{\theta}{2} & 1 & -1\end{array}\right|$ lies in the interval

If for a matrix $A,{A^2} + I = O,$ where $I$ is the identity matrix, then $A$ equals

The matrices $P = \left[ {\begin{array}{cccccccccccccccccccc} {{u_1}}&{{v_1}}&{{w_1}}\\ {{u_2}}&{{v_2}}&{{w_2}}\\ {{u_3}}&{{v_3}}&{{w_3}} \end{array}} \right];Q = \frac{1}{9}\left[ {\begin{array}{cccccccccccccccccccc} 2&2&1\\ {13}&{ - 5}&m\\ { - 8}&1&5 \end{array}} \right]$ are such that $PQ = I$, an identity matrix. Solving the equation $\left[ {\begin{array}{cccccccccccccccccccc} {{u_1}}&{{v_1}}&{{w_1}}\\ {{u_2}}&{{v_2}}&{{w_2}}\\ {{u_3}}&{{v_3}}&{{w_3}} \end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc} x\\ y\\ z \end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc} 1\\ 1\\ 5 \end{array}} \right]$, the value of $y$ comes out to be $- 3$. Then, the value of $m$ is equal to

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations $x + y + z = 5,\,\,x + 2y + 3z = \mu$ and $x + 3y + \lambda z = 1$ is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution, then

Let $A, B, C$ be three square matrices of the same order, such that whenever $A B=A C$ then $B=C$, if $A$ is

$\left| {\begin{array}{cccccccccccccccccccc} {a - b}&{b - c}&{c - a}\\ {x - y}&{y - z}&{z - x}\\ {p - q}&{p - r}&{r - p} \end{array}} \right|$ is equal to

If ${A}^n=\left[\begin{array}{ccc}2^n & 0 & 0 \\ 0 & 2^n & 0 \\ 0 & 0 & 2^n\end{array}\right]$, then ${A}^{10}=$

Which one of the following statement is true?

If the system of linear equations.
$\begin{gathered} 8x + y + 4z = - 2 \quad \\ x + y + z = 0 \quad \\ \lambda x - 3y = \mu \quad \\ \end{gathered}$
has infinitely many solutions, then the distance of the point $\left(\lambda, \mu,-\frac{1}{2}\right)$ from the plane $8 x+y+4 z+2=0$ is

Consider the following system of equations
$x + 2y - 3z = a$
$2x + 6y - 11z = b$
$x - 2y + 7z = c$
Where, $a,\,b$ and $c$ are real constants. Then, the system of equations

The matrix $A=\left(\begin{array}{cc}p & -q \\ q & p\end{array}\right)$ is orthogonal if and only if

If
$A=\left[\begin{array}{ll} \alpha & 2 \\ 2 & \alpha \end{array}\right] \text { and }\left|A^3\right|=125$
then $\alpha$ is equal to

If the system of equations

$\begin{array}{l} x+y+a z=b \\ 2 x+5 y+2 z=6 \\ x+2 y+3 z=3 \end{array}$

has infinitely many solutions, then $2 a+3 b$ is equal to

If $\mathrm{P}=\left[\begin{array}{cc}\sin \theta & \cos \theta \\ -\cos \theta & \sin \theta\end{array}\right]$, then $\mathrm{P}^{-1}=$

Which of the given values of $\mathrm{x}$ and $y$ make the following pairs of matrices equal:
$\left[\begin{array}{cc} 3 x+7 & 5 \\ y+1 & 2-3 x \end{array}\right] \cdot\left[\begin{array}{cc} 0 & y-2 \\ 8 & 4 \end{array}\right]$

In how many ways, the determinant of order 3 can be expanded?

If $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ and $A^2=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]$, then

Let $A$ be a $3 \times 3$ matrix such that $A^2-5 A+7 I=O$.

Statement $-I: A^{-1}=\frac{1}{7}(5 I-A)$.

Statement $-I I$ : The polynomial $A^3-2 A^2-3 A+I$ can be reduced to $5(A-4 I)$. Then

The value of $\left| {\begin{array}{cccccccccccccccccccc} {^{10}{C_4}}&{^{10}{C_5}}&{^{11}{C_m}}\\ {^{11}{C_6}}&{^{11}{C_7}}&{^{12}{C_{m + 2}}}\\ {^{12}{C_8}}&{^{12}{C_9}}&{^{13}{C_{m + 4}}} \end{array}} \right| = 0,$ when $m$ is equal to