Matrices practice · Mixed difficulty

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

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {3\;\;3\;\;3}\\ {3\;\;3\;\;3}\\ {3\;\;3\;\;3} \end{array}} \right],\;{A^4}\;$ is equal to

If $\left(1+a x+b x^2\right)^4=a_0+a_1 x+a_2 x^2+\ldots+a_8 x^8$, where $a, b, a_0, a_1, \ldots, a_8 \in R$ such that $a_0+a_1+a_2 \neq 0$ and $\left|\begin{array}{lll}a_0 & a_1 & a_2 \\ a_1 & a_2 & a_0 \\ a_2 & a_0 & a_1\end{array}\right|=0$, then

Let $A$ and $B$ are two square matrices such that $AB = A\;$ and $\;BA = B,\;$ then ${A^2}$ equals to

If a matrix has 8 elements, then which of the following will not be a possible order of the matrix?

Let $a,\;b,\;c$ are positive real numbers. The following system of equations $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1, - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$ in $x,y\;$ and $z$ has

If $\theta, \phi \,\varepsilon \,{R}$, then the determinant $\Delta=\left|\begin{array}{ccc} \cos \theta & -\sin \theta & 1 \\ \sin \theta & \cos \theta & 1 \\ \cos (\theta+\phi) & -\sin (\theta+\phi) & 0 \end{array}\right|$ lies in the interval

Let $A = \left[ {\begin{array}{cccccccccccccccccccc} {1\;\;2}\\ {3\;\;4} \end{array}} \right]$ and $\;B = \left[ {\begin{array}{cccccccccccccccccccc} {\;a\;\;0}\\ {0\;\;b} \end{array}} \right]$, $a,b\;N.$ then

If $A$ and $B$ are square matrices of size $n \times n$ such that ${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$ then which of the following will be always true ?

If $A=\left[\begin{array}{cc}0 & -i \\ i & 0\end{array}\right], B=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$, then $A B+B A$ is

If $A$ is non-singular and $(A-2 I)(A-4 I)=O$, then $\frac{1}{6} A+\frac{4}{3} A^{-1}=$

The value of $\left| {\begin{array}{cccccccccccccccccccc} x&p&q\\ p&x&q\\ p&q&x \end{array}} \right|$ is

The integer represented by the determinant $\left[ {\begin{array}{cccccccccccccccccccc} {215}&{342}&{511}\\ 6&7&8\\ {36}&{49}&{54} \end{array}} \right]$ is exactly divisible by

If $A=\left[a_{i j}\right]_{4 \times 4}$, such that $a_{i j}=\left\{\begin{array}{l}2, \text { when } i=j \\ 0, \text { when } i \neq j\end{array}\right.$, then $\left\{\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj} A))}{7}\right\}$ is [when $\{\cdot\}$ represents fractional part function]

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 2&4&5\\ 4&8&{10}\\ { - 6}&{ - 12}&{ - 15} \end{array}} \right],\;$ the rank of $A$ is equal to

If $A^5=O$ such that $A^n \neq I$ for $1 \leq n \leq 4$, then $(I-A)^{-1}$ is equal to

If $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$ and $P=B^T A B$ then $B P^{2016} B^T$ is equal to

Let $B=\left[\begin{array}{rrr}2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1\end{array}\right]$. If the product $B A$ has inverse $C=\left[\begin{array}{rrr}-1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2\end{array}\right]$, then $A^{-1}$ equals

Suppose $A$ is square matrix such that $A^3=I$, then $(A+I)^3+(A-I)^3-6 A$ equals

There are two numbers $x$ making the value of the determinant $\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & \mathrm{x} & -1 \\ 0 & 4 & 2 \mathrm{x}\end{array}\right|$ equal to 86 . The sum of these two numbers, is

Let $f\left( \alpha \right) = \left[ {\begin{array}{cccccccccccccccccccc} {\cos \alpha }&{ - \sin \alpha }&0\\ {\sin \alpha }&{\cos \alpha }&0\\ 0&0&1 \end{array}} \right],$ where $\alpha \in R.$ Then, ${\left( {F\left( \alpha \right)} \right)^{ - 1}}$ is equal to

The number of the solutions of the equation $\log \left(x^2-\right.$ $6 x+7)=\log (x-3)$ is

Let ${A}$ be a $2 \times 2$ matrix

Assertion: $\operatorname{adj}(\operatorname{adj} A)=A$

Reason: $|\operatorname{adj} {A}|=|{A}|$

Let ${N}$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations

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

has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is

If $\mathrm{A}$ and $\mathrm{B}$ are two matrices such that $\mathrm{A}+\mathrm{B}$ and $\mathrm{AB}$ are both defined, then

Matrix $A$ such that ${A^2} = 2A - I$, where $I$ is the identity matrix. Then, for $n \ge 2,\;{A^n}$ is equal to

The number of values of $k$ for which the linear equations $4x + ky + 2z = 0,\,\,\,kx + 4y + z = 0,\,\,2x + 2y + z = 0$ Possess a non - zero solutions is

If $\Delta_r=\left|\begin{array}{ccc}r & x & \frac{n(n+1)}{2} \\ 2 r-1 & y & n^2 \\ 3 r-2 & z & \frac{n(3 n-1)}{2}\end{array}\right|,$ then $\sum\limits_{r = 1}^n {{\Delta _r}}$ is equal to

Let $A=\left[\begin{array}{ll}1 & 4 \\ 3 & 2\end{array}\right].$ If $\theta$ is the angle between the two non-zero column vectors $X$ such that $A X=\lambda X$ for some scalar $\lambda,$ then $\tan \theta$ is equal to

$A$ is a square matrix of order 4 and $I$ is a unit matrix, then it is true that