Matrices practice · Easy

If $\left| {\begin{array}{cccccccccccccccccccc} x&{{x^2}}&{1 + {x^3}}\\ y&{{y^2}}&{1 + {y^3}}\\ z&{{z^2}}&{1 + {z^3}} \end{array}} \right| = 0$ and $x,y,z$ are all distinct, then $xyz$ is equal to

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

If
$\left(\begin{array}{cc} 1 & -\tan \theta \\ \tan \theta & 1 \end{array}\right)\left(\begin{array}{cc} 1 & \tan \theta \\ -\tan \theta & 1 \end{array}\right)^{-1}=\left[\begin{array}{cc} a & -b \\ b & a \end{array}\right],$ then

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

The system of equations
$\begin{array}{l} kx + y + z = 1\\ x + ky + z = k\\ x + y + kz = {k^2} \end{array}$
have no solution, if $k$ equals

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 $A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$, then $A^8$ equals

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 $\frac{1}{6} A+\frac{4}{3} A^{-1}=$

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

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

Let for $A=\left[\begin{array}{lll}1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2\end{array}\right],|A|=2$. If $|2 \operatorname{adj}(2 \operatorname{adj}(2 A))|$ $=32^{n}$, then $3 n+\alpha$ is equal to

Let $A_k=\left[\begin{array}{cc} \cos ^2 \alpha_k & \cos \alpha_k \sin \alpha_k \\ \cos \alpha_k \sin \alpha_k & \sin ^2 \alpha_k \end{array}\right] \quad(k=1,2)$
If the difference between $\alpha_1$ and $\alpha_2$ is an odd multiple of $\pi / 2$, then $A_1 A_2$ is a matrix whose sum of all its elements is equal to

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

If $\omega \neq 1$ is a cube root of unity, find the value of $\left|\begin{array}{cccc}1 & \omega & \omega^2 & \omega^3 \\ \omega & \omega^2 & \omega^3 & 1 \\ \omega^2 & \omega^3 & 1 & \omega \\ \omega^3 & 1 & \omega & \omega^2\end{array}\right|$

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