Matrices practice · Easy

If $A$ is a symmetric matrix and $n\; \in \;N$, then ${A^n}\;$ is

If $A$ is a skew-symmetric matrix of odd order, then $\left| {\mathrm{adj}\;A} \right|$ is equal to

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {1\,\,\,\; - 2}\\ {4\,\,\,\;\;5} \end{array}} \right]\mathrm{\;and\;}f\left( t \right) = {t^2} - 3t + 7,\mathrm{\;then\;}f\left( A \right) + \left[ {\begin{array}{cccccccccccccccccccc} {3\;\;\;\;\;\;\;\;6}\\ { - 12\,\,\,\,\; - 9} \end{array}} \right]\mathrm{\;}$ is equal to

If $A=\left[\begin{array}{cc}a b & b^2 \\ -a^2 & -a b\end{array}\right]$, then $A^2$ is equal

Let the system of linear equation $x + 2y + z = 2,\alpha x + 3y - z = \alpha , - \alpha x + y + 2z = - \alpha$ be inconsistent. Then $\alpha$ is equal to:

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

$\left| {\begin{array}{cccccccccccccccccccc} {1 + {{\sin }^2}\mathrm{\theta }}&{{{\sin }^2}\mathrm{\theta }}&{{{\sin }^2}\mathrm{\theta }}\\ {{{\cos }^2}\mathrm{\theta }}&{1 + {{\cos }^2}\mathrm{\theta }}&{{{\cos }^2}\mathrm{\theta }}\\ {4\sin 4\mathrm{\theta }}&{4\sin 4\mathrm{\theta }}&{1 + 4\sin 4\mathrm{\theta }} \end{array}} \right| = 0,$ then $\sin 4\theta$ equals to

The value of $x$ for which the matrix $A=\left[\begin{array}{ccc}2 / x & -1 & 2 \\ 1 & x & 2 x^2 \\ 1 & 1 / x & 2\end{array}\right]$ is singular is

If $\left[\begin{array}{cc}x+y & 2 x+z \\ x-y & 2 z+w\end{array}\right]=\left[\begin{array}{cc}4 & 7 \\ 0 & 10\end{array}\right]$, then values of $x, y, z$, ware

If
$A=\left[\begin{array}{cc} 2 x & 0 \\ x & x \end{array}\right] \text { and } A^{-1}=\left[\begin{array}{rr} 1 & 0 \\ -1 & 2 \end{array}\right]$
then $x$ is equal to

Assertion : The matrix $A=\left[\begin{array}{lll}9 & 1 & 2 \\ 3 & 7 & 4\end{array}\right]$ does not possesses any inverse.

Reason : ${A}$ is not a square matrix.

If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1\end{array}\right]$ and $A^{-1}=\left[\begin{array}{ccc} 1 / 2 & -1 / 2 & 1 / 2 \\ -4 & 3 & y \\ 5 / 2 & -3 / 2 & 1 / 2 \end{array}\right], \text { then}$

$x + ky - z = 0,\;3x - ky - z = 0\;$ and $x - 3y + z = 0\;$ has non-zero solution for $k$ is equal to

If $A-2 B=\left[\begin{array}{ll}1 & 5 \\ 3 & 7\end{array}\right]$ and $2 A-3 B=\left[\begin{array}{rr}-2 & 5 \\ 0 & 7\end{array}\right]$, then matrix $B$ is equal to

If $\mathrm{P}=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right)$ then, $\operatorname{det}(\operatorname{Adj} P)=$

If $A=\left[a_{i j}\right]_{4 \times 4}$ such that $a_{i j}=\left\{\begin{array}{ll}2, & i=j \\ 0, & i \neq j\end{array}\right.$, then $\left\{\frac{1}{7} \operatorname{det}(\operatorname{adj}(\operatorname{adj} A)\}\right.$ is, where {.} denotes the fractional part function.

If $A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]$, then $|A||\operatorname{adj} A|$ is equal to

If $A=\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$, then $A^T+A=I_2$, if

If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A$, then $A^2+B^2$ is equal to

If $A=\left[\begin{array}{rr}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta\end{array}\right], B=\left[\begin{array}{rr}1 & 0 \\ -1 & 1\end{array}\right], C=A B A^T$, then $A^T C^n A$ equals to $\left(n \in I^{+}\right)$

If $A$ is a $2 \times 2$ matrix and $I$ is an identity matrix of order $2 \,\&\,|A-\lambda I|=0$ gives value of $\lambda$ as $-1 \,\&\, 3$ then trace of $A^2$ is equal to ____

If $P = \left[ {\begin{array}{cccccccccccccccccccc} 1&2&3\\ 2&3&4\\ 3&4&5 \end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc} { - 1}&{ - 2}\\ { - 2}&0\\ 0&{ - 4} \end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc} { - 4}&{ - 5}&{ - 6}\\ 0&0&1 \end{array}} \right]$, then ${P_{22}}$ is equal to

If $A=\left[\begin{array}{cc}1 & 2 \\ 2 & 1\end{array}\right]$ and $f(x)=\frac{1+x}{1-x}$, then $f(A)$ is

If ${\Delta _1} = \left| {\begin{array}{cccccccccccccccccccc} x&{\sin \theta }&{\cos \theta }\\ { - \sin \theta }&{ - x}&1\\ {\cos \theta }&1&x \end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{cccccccccccccccccccc} x&{\sin 2\theta }&{\cos 2\theta }\\ { - \sin 2\theta }&{ - x}&1\\ {\cos 2\theta }&1&x \end{array}} \right|,\,\,x \ne 0,$ then for all $\theta \in \left( {0,\frac{\pi }{2}} \right)$

If $A$ is a non-singular matrix of order 3 , then $|\operatorname{adj} A|=|A|^{\mathrm{n}}$. Here the value of $n$ is

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

If $A$ is square matrix of order $n, a=$ maximum number of distinct entries. If $A$ is triangular matrix, $b=$ maximum number of distinct entries. If $A$ is a diagonal matrix. $c=$ minimum number of zeros. If $A$ is a triangular matrix if $a+5=c+2 b$

If $a + x = b + y = c + z + 1,\,$ where $a,\,b,\,c,\,x,\,y,\,z$ are non-zero distinct real numbers, then $\left| {\begin{array}{cccccccccccccccccccc} x&{a + y}&{x + a}\\ y&{b + y}&{y + b}\\ z&{c + y}&{z + c} \end{array}} \right|$ is equal to

If $A=\left[\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]$, then

If a square matrix $A$ is such that $A{A^T} = I = {A^T}A\,\,\,\mathrm{then}\,\,\,\,\left| A \right|$ then is equal to