Matrices practice · Easy

If $A$ is a skew Hermitian matrix, then the main diagonal elements of $A$ are all

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

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

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

Let $S_k=\left(\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right), k \in \mathbf{N}$. Then $\left(S_2\right)^n\left(S_k\right)^{-1}$ (where $n \in \mathbf{N}$ ) 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 $A = \left( {\begin{array}{cccccccccccccccccccc} 2&2\\ 9&4 \end{array}} \right)\,\,\mathrm{and}\,\,I = \left( {\begin{array}{cccccccccccccccccccc} 1&0\\ 0&1 \end{array}} \right)$ 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

If $A=\left[\begin{array}{cc}4 & 2 \\ -1 & 1\end{array}\right]$, then the value of $(A-2 I)(A-3 I)$ is

If $c < 1$ and the system of equations $x + y - 1 = 0,2x - y - c = 0$ and $- bx + 3by - c =$ is consistent, then the possible real values of $b$ are

the system of simultaneous equations
$\begin{array}{l} kx + 2y - z = 1\\ \left( {k - 1} \right)y - 2z = 2\\ \left( {k + 2} \right)z = 3 \end{array}$
Have a unique solution if $k$ equals

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:

If $A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]$ and $a^2+b^2+c^2+d^2=1$, then $A^{-1}$ 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:

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 :

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 $A=\left[\begin{array}{cc}\log x & -1 \\ -\log x & 2\end{array}\right]$ and if $\operatorname{det}(A)=2$, then the value of $x$ is equal to

If $A$ is a skew symmetric matrix of order $n$ and $C$ is a column matrix of order $n \times 1,\;\mathrm{then}\;{C^T}AC\;$ is

If $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ such that $a d-b c \neq 0$, then $A^{-1}$, is

If $A$ is an idempotent matrix i.e., $A^2=A$ and $B=I-A$, then which of the following is not true?