Matrices practice · Mixed difficulty

The system of linear equations
$\begin{array}{l} \lambda x + 2y + 2z = 5\\ 2\lambda x + 3y + 5z = 8\\ 4x + \lambda y + 6z = 10\,\,\mathrm{has} \end{array}$

Let the determinant of a $3 \times 3$ matrix $A$ be 6, then $B$ is a matrix defined by $B = 5{A^2}$. Then, determinant of $B$ is

If $f(x)=a x^2+b x+c, a, b, c \in R$ and equation $f(x)-x=0$ has imaginary roots $\alpha, \beta, \gamma$ and $\delta$ be the roots of $f(f(x))-x=0$, then $\left|\begin{array}{ccc}2 & \alpha & \delta \\ \beta & 0 & \alpha \\ \gamma & \beta & 1\end{array}\right|$ is

Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix, where $a_{i j}= \begin{cases}1, & \text { if } i=j \\ -x & \text { if }|i-j|=1 \\ 2 x+1, & \text { otherwise }\end{cases}$ Let a function be defined as $f(x)=\operatorname{det}(A)$. Then the sum of maximum and minimum values of $f$ on ${R}$ is equal to

If $A=\left[\begin{array}{lll}a^2 & a b & a c \\ a b & b^2 & b c \\ a c & b c & c^2\end{array}\right]$ and $B=\left[\begin{array}{ccc}0 & c & -b \\ -c & 0 & a \\ b & -a & 0\end{array}\right]$ then the product $A B$ is equal to

The number of solutions of the matrix equation $X^2=\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right]$ is

If $A$ and $B$ are square matrices such that $A^*=A$ and $B^*=B$, where $A^*$ denotes the conjugate transpose of $A$, then $(A B-B A)^*=$

In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then, it can be decomposed into $n$ determinant, where $n$ has the value

If $A = \left[ {\begin{array}{cccccccccccccccccccc} { - 1}&2\\ 2&{ - 1} \end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc} 3\\ 1 \end{array}} \right],\;Ax = B$, then $X$ is equal to

A matrix of order $3 \times 3$ are formed by using the elements of the set $A=\{-3,-2,-1,0,1,2,3\}$, then find the probability that the matrix is either symmetric of skew symmetric is

The determinant $\Delta=\left|\begin{array}{ccc} 1 & \cos (\beta-\alpha) & \cos (\gamma-\alpha) \\ \cos (\alpha-\beta) & 1 & \cos (\gamma-\beta) \\ \cos (\alpha-\gamma) & \cos (\beta-\gamma) & 1 \end{array}\right| \text { is equal to }$

From the matrix equation $AB = AC$ we can conclude $B = C$ provided that

The value of $\Delta=\left|\begin{array}{lll}1 & 1+a c & 1+b c \\ 1 & 1+a d & 1+b d \\ 1 & 1+ae & 1+b e\end{array}\right|,$ is

The value of $a$ and $b$, for which the system of equations
$2x + 3y + 6z = 8,$ $x + 2y + az = 5$, $3x + 5y + 9z = b$ has no solution, are

If $\alpha, \beta \neq 0$ and $f(n)=\alpha^n+\beta^n$ and $\left|\begin{array}{ccc}3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4)\end{array}\right|=k(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2$, then $k$ is equal to

If $\left| {\begin{array}{cccccccccccccccccccc} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&{{b_2}}&{{c_2}}\\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right| = 5,$ then the value of $\left| {\begin{array}{cccccccccccccccccccc} {{b_2}{c_3} - {b_3}{c_2}}&{{c_2}{a_3} - {c_3}{a_2}}&{{a_2}{b_3} - {c_3}{b_2}}\\ {{b_3}{c_1} - {b_1}{c_3}}&{{c_3}{a_1} - {c_1}{a_3}}&{{a_3}{b_1} - {a_1}{b_3}}\\ {{b_1}{c_2} - {b_2}{c_1}}&{{c_1}{a_2} - {c_2}{a_1}}&{{a_1}{b_2} - {a_2}{b_1}} \end{array}} \right|$ is

If $A = \left( {\begin{array}{cccccccccccccccccccc} 2&2\\ 9&4 \end{array}} \right)\,\,\mathrm{and}\,\,I = \left( {\begin{array}{cccccccccccccccccccc} 1&0\\ 0&1 \end{array}} \right)$ the $10\,{A^{ - 1}}$ is equal to

A square non-singular matrix $A$ satisfies $A^2-A+2 I=0$, then $A^{-1}=$

If $A$ is an orthogonal matrix, then

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {1\;\,\,\, - 3}\\ {2\;\;\;\;\;k} \end{array}} \right]\;$ and $\;{A^2} - 4A = 10I = A\;$ then $k$ is equal to

Let $S_k=\left(\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right), k \in {N}$. Then $\left(S_2\right)^n\left(S_k\right)^{-1}$ (where $n \in {N}$ ) is equal to:

Let $A$ be a $3 \times 3$ real matrix such that $A\left( {\begin{array}{cccccccccccccccccccc} 1 \\ 1 \\ 0 \end{array}} \right) = \left( {\begin{array}{cccccccccccccccccccc} 1 \\ 1 \\ 0 \end{array}} \right);A\left( {\begin{array}{cccccccccccccccccccc} 1 \\ 0 \\ 1 \end{array}} \right) = \left( {\begin{array}{cccccccccccccccccccc} { - 1} \\ 0 \\ 1 \end{array}} \right)\,\,{\text{and}}\,\,A\left( {\begin{array}{cccccccccccccccccccc} 0 \\ 0 \\ 1 \end{array}} \right) = \left( {\begin{array}{cccccccccccccccccccc} 1 \\ 1 \\ 2 \end{array}} \right).$ If $X = {({x_1},{x_2},{x_3})^T}$ and $I$ is an identity matrix of order 3, then the system $(A - 2I)X = \left( {\begin{array}{cccccccccccccccccccc} 4 \\ 1 \\ 1 \end{array}} \right)$ has:

Let matrix $A=\left[\begin{array}{ccc}x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2\end{array}\right]$ where $x, y, z \in N$. If $\operatorname{det}(\operatorname{adj}(\operatorname{adj} A))=2^8 \cdot 3^4$ then the number of such matrices $A$ is

[Note: $\operatorname{adj} A$ denotes adjoint of square matrix $A$]

For two $3 \times 3$ matrices $A$ and $B$, let $A+B=2 B^{\mathrm{T}}$ and $3 \mathrm{A}+2 \mathrm{B}=$ $\mathrm{I}_{3}$, where $\mathrm{B}^{\mathrm{T}}$ is the transpose of $\mathrm{B}$ and $\mathrm{I}_{3}$ is $3 \times 3$ identity matrix. Then :

Let $A=\left(\begin{array}{ll}m & n \\ p & q\end{array}\right), d=|A| \neq 0$ and $|A-d(\operatorname{Adj} A)|=0$. Then

If $\omega \neq 1$ is the complex cube root of unity and matrix $H=\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]$, then $\mathrm{H}^{70}$ is equal to

For positive numbers $x,y$ and $z$, the numerical value of the determinant $\left| {\begin{array}{cccccccccccccccccccc} 1&{{{\log }_x}y}&{{{\log }_x}z}\\ {{{\log }_y}x}&1&{{{\log }_y}z}\\ {{{\log }_z}x}&{{{\log }_z}y}&1 \end{array}} \right|$ is

Consider the following system of equations

$\begin{array}{l} \alpha x+2 y+z=1 \\ 2 \alpha x+3 y+z=1 \\ 3 x+a y+2 z=b \end{array}$

For some $\alpha, \beta \in \mathbb{R}$. then which of the following is NOT correct?

If $\left| {\begin{array}{cccccccccccccccccccc} x&3&6\\ 3&6&x\\ 6&x&3 \end{array}} \right| = \left| {\begin{array}{cccccccccccccccccccc} 2&x&7\\ x&7&2\\ 7&2&x \end{array}} \right| = \left[ {\begin{array}{cccccccccccccccccccc} 4&5&x\\ 5&x&4\\ x&4&5 \end{array}} \right] = 0,$ then $x$ is equal to

If $A$ and $B$ are symmetric matrices of order $n(A \neq B)$, then