Matrices practice · Mixed difficulty

If $a_1, a_2, a_3, a_5$ are consecutive terms of arithmetic progression with common difference 3, then the value of $\left|\begin{array}{lll}a_3^2 & a_2 & a_1 \\ a_4^2 & a_3 & a_2 \\ a_5^2 & a_4 & a_3\end{array}\right|$ is

The number of values of $k$ for which the system of equations: $(k + 1)x + 8y = 4k,kx + (k + 3)y = 3k - 1$ has solution is

Consider the system of equations in $x, y, z$ as
$x \sin 3 \theta-y+z=0$

$x \cos 2 \theta+4 y+3 z=0$

$2 x+7 y+7 z=0$
If this system has a non-trivial solution, then for any integer $n$, values of $\theta$ are given

There are two possible values of $A$ in the solution of the matrix equation $\left[\begin{array}{rr}2 A+1 & -5 \\ -4 & A\end{array}\right]^{-1}\left[\begin{array}{rr}A-5 & B \\ 2 A-2 & C\end{array}\right]=\left[\begin{array}{rr}14 & D \\ E & F\end{array}\right],$ where $A, B, C, D, E, F$ are real numbers. The absolute value of the difference of these two solutions, is

Consider the following statements

I. If ${AB}$ and ${BA}$ are both defined, then they must be equal i.e., $A B=B A$.

II. If ${AB}$ and ${BA}$ are both defined, it is not necessary that ${AB}={BA}$.

Choose the correct option.

If $A$ is a square matrix of order 3 such that $A^2=$ $2 A$, then $|A|^2$ is equal to

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {1\;\;\;\;\;\;\;x}\\ {{x^2}\;\;\;4y} \end{array}} \right]$ and $\;B = \left[ {\begin{array}{cccccccccccccccccccc} { - 3\;\;\;\;\;1}\\ {1\;\;\;\;\;\;0} \end{array}} \right],\;\mathrm{adj}\;A + B = \left[ {\begin{array}{cccccccccccccccccccc} {1\;\;\;\;0}\\ {0\;\;\;\;\;1} \end{array}} \right]$, Then values of $x$ and $y$ are

Let $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&2\\ { - 5}&1 \end{array}} \right]\mathrm{and}\;{A^{ - 1}} = xA + yI$, then the value of $x$ and $y$ are

Let for any matrix $M,\;{M^{ - 1}}\;$ exist which of the following is not true?

If $x^a y^b=e^m, x^c y^d=e^n, \Delta_1=\left|\begin{array}{ll}m & b \\ n & d\end{array}\right|,$ $\Delta_2=\left|\begin{array}{ll}a & m \\ c & n\end{array}\right|$ and $\Delta_3=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|,$ then the values of $x$ and $y$ are

The system of equations kx + y+ z = 1, x+ ky + z = k and x +y +zk = ${k^2}$ has no solution, If k is equal to

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 3&4\\ 5&7 \end{array}} \right]$ then $A.\left( {\mathrm{adj}\;A} \right)$ is equal to

If $a, b, c$ are positive integers such that $a > b > c$ and $\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{array}\right|=-2$ then $3 a+7 b-10 c$ equals

The inverse of a skew-symmetric matrix (if it exists) is

Let $A=\left(a_{i j}\right)_{3 \times 3}$ be a matrix with $a_{i j} \in {C}$. Let $B$ be a matrix obtained by interchanging two columns of $A.$ Then $\operatorname{det}(A+B)$ is equal to

If $A$ is a square matrix then which one of the following is not a symmetric matrix

If $A$ is an invertible matrix of order $n$, then the determinant of $adj\left( A \right)$ is equal to

If the trivial solution is the only solution of the system of equations $x - ky + z = 0,\,\,kx + 3y - kz = 0,\,\,\,3x + y - z = 0$ Then, the set of values of $k$ is

Let $A\left[\begin{array}{cc}1 & 2 \\ -5 & 1\end{array}\right]$ and $A^{-1}=x A+y I$, then the value of $(2 x+3 y)$ is______

If $A$ is a singular matrix, then adj $A$ is

If $\left[ {\begin{array}{cccccccccccccccccccc} 1&2&x\\ 4&{ - 1}&7\\ 2&4&{ - 6} \end{array}} \right]$ is a singular matrix, then $x$ is equal to

If $\left| {\begin{array}{cccccccccccccccccccc} x&y&z\\ { - x}&y&z\\ x&{ - y}&z \end{array}} \right| = kxyz,$ then $k$ is equal to

If $\omega$ is a complex cube root of unity, then the value of $\left| {\begin{array}{cccccccccccccccccccc} {x + 1}&\omega &{{\omega ^2}}\\ \omega &{x + {\omega ^2}}&1\\ {{\omega ^2}}&1&{x + \omega } \end{array}} \right|$ is

If $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$, then $A^2$ is equal to

The value of $\Delta=\left|\begin{array}{lll}\left(a^x+a^{-x}\right)^2 & \left(a^x-a^{-x}\right)^2 & 1 \\ \left(a^y+a^{-y}\right)^2 & \left(a^y-a^{-y}\right)^2 & 1 \\ \left(a^z+a^{-z}\right)^2 & \left(a^z-a^{-z}\right)^2 & 1\end{array}\right|,$ is

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations

$\begin{aligned} &2 x-y+2 z=2 \\ &x-2 y+\lambda z=-4 \\ &x+\lambda y+z=4 \end{aligned}$

has no solution, than the set $S$

In a square matrix $A$ of order 3 the elements $a_{i i}$ 's are the sum of the roots of the equation $x^2-(a+b) x+a b=0;$ $a_{i, \,i+1}$ 's are the product of the roots, $a_{i, \,i-1}$ 's are all unity and the rest of the elements are all zero.

The value of the $\operatorname{det}(A)$ is equal to

Suppose $a \in {R}$. Let $f(x)=\left|\begin{array}{ccc} x+a & x & x \\ x & x+a & x \\ x & x & x+a \end{array}\right|$ Then $f(2 x)-f(x)$ is equal to

If $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]$ then $A^3-r A^2-q A=$

If $A$ and $B$ are two square matrices such that $B=-A^{-1} B A,$ then $(A+B)^2$ is equal to