Matrices practice · Easy

If square matrices $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{AB}=\mathrm{A}$ and $\mathrm{BA}=\mathrm{B}$, then $\mathrm{B}^2=$

If $A$ is a square matrix, then $\operatorname{adj} A^{\top}-(\operatorname{adj} A)^T$ is equal to

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

Statement $-1: \operatorname{adj}(\operatorname{adj} \mathrm{A})=\mathrm{A}$

Statement -2 : $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|$

The restrictions on $n, k$ and $p$ so that $P Y+W Y$ will be defined are

If $\mathrm{A}=\left[\begin{array}{cc}5 \mathrm{a} & -\mathrm{b} \\ 3 & 2\end{array}\right]$ and $\mathrm{A}$ adj $\mathrm{A}=\mathrm{AA}^{\mathrm{T}}$, then $5 \mathrm{a}+\mathrm{b}$ is equal to:

In the determinant $\left| {\begin{array}{cccccccccccccccccccc} 0&1&{ - 2}\\ { - 1}&0&3\\ 2&{ - 3}&0 \end{array}} \right|$, the value of cofactor to its minor of the element $- 3$ is

If $A$ is a square matrix, then $\operatorname{adj} A^{\top}-(\operatorname{adj} A)^{\top}$ is equal to

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

Statement-1 $\operatorname{adj}(\operatorname{adj} A)=A$

Statement-2 $|\operatorname{adj} A|=|A|$

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&0 \\ 1&1 \end{array}} \right]$ and $I = \left[ {\begin{array}{cccccccccccccccccccc} 1&0 \\ 0&1 \end{array}} \right],$ then which one of the following holds for all $n \geqslant 1,$ by the principle of mathematical induction

The number of non-zero diagonal matrices of order 4 satisfying $A^2=A$ is

Let $A$ and $B$ are symmetric matrices of order 3.

Statement-1 $A(B A)$ and $(A B) A$ are symmetric matrices.

Statement-2 $A B$ is symmetric matrix, if matrix multiplication of $A$ with $B$ is commutative.

If $A=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3\end{array}\right], B=$ adj $A$ and $C=3 A$, then $\frac{|\operatorname{adj} B|}{|C|}$ is equal to

If $A$ and $B$ are any two different square matrices of order $n$ with $A^3=B^3$ and $A(A B)=B(B A)$, then

If $\mathrm{A}$ is a square matrix for which $a_{i j}=i^2-j^2$, then $\mathrm{A}$ is

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

If $f\left( \alpha \right) = \left| {\begin{array}{cccccccccccccccccccc} 1&\alpha &{{\alpha ^2}}\\ \alpha &{{\alpha ^2}}&1\\ {{\alpha ^2}}&1&\alpha \end{array}} \right|$, then $f\left( {\sqrt[3]{3}} \right)$ is equal to

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&{ - 1}&1\\ 2&1&{ - 3}\\ 1&1&1 \end{array}} \right],\;10\;B = \left[ {\begin{array}{cccccccccccccccccccc} 4&2&2\\ { - 5}&0&\alpha \\ 1&{ - 2}&3 \end{array}} \right]$ and $B$ is the inverse of $A$, then the value of $\alpha$ is

If $A_\alpha=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$, then $A_\alpha A_\beta$ is equal to

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {2\;\;1\;\;0}\\ {0\;\;2\;\;1}\\ {1\;\;0\;\;2} \end{array}} \right],\;$ then $|adj\,A|$ is equal to

If ${P}$ is orthogonal matrix, ${A}$ is involutary matrix and $Q=P A P^T, X=P^T Q^{1000} P$ then $X^{-1}$ is

The determinant $\left|\begin{array}{ccc}y^2 & -x y & x^2 \\ a & b & c \\ a^{\prime} & b^{\prime} & c^{\prime}\end{array}\right|$ is equal to

If $A=\left[\begin{array}{lll}1-2 & 3\end{array}\right], B=\left[\begin{array}{r}2 \\ -3 \\ -1\end{array}\right]$, then $A B$ is equal to

The characteristic roots of the matrix $\left[ {\begin{array}{cccccccccccccccccccc} 1&0&0\\ 2&3&0\\ 4&5&6 \end{array}} \right]$ are

If
$\left[\begin{array}{cc} 3 & -4 \\ 1 & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 3 \\ 11 \end{array}\right]$
then

If $\left| {\begin{array}{cccccccccccccccccccc} {1 + a}&1&1\\ {1 + b}&{1 + 2b}&1\\ {1 + c}&{1 + c}&{1 + 3c} \end{array}} \right| = 0$, where $a \ne 0,b \ne 0,c \ne 0,$ then ${a^{ - 1}} + {b^{ - 1}} + {c^{ - 1}}$

If for the matrix $A, A^3=I$, then $A^{-1}=$

If $a, b$ and $c$ are complex numbers, the determinant
$\Delta = \left| {\begin{array}{cccccccccccccccccccc} 0&{ - b}&{ - c} \\ {\bar b}&0&{ - a} \\ {\bar c}&{\bar a}&0 \end{array}} \right|{\text{ }}\,{\text{is }}$

$A=\left[\begin{array}{ccc}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{array}\right]$ is a matrix satisfying the equation $A A^T=9 I$, where $I$ is $3 \times 3$ matrix, then the ordered pair $(a, b)$ is equal to

Let m and M be respectively the minimum and maximum values of $\left| {\begin{array}{cccccccccccccccccccc} {{{\cos }^2}x}&{1 + {{\sin }^2}x}&{\sin 2x}\\ {1 + {{\cos }^2}x}&{{{\sin }^2}x}&{\sin 2x}\\ {{{\cos }^2}x}&{{{\sin }^2}x}&{1 + \sin 2x} \end{array}} \right|.$ Then, the ordered pair $(m,M)$ is equal to

Consider the following statements

I. To every rectangular matrix $A=\left[a_{i j}\right]$ of order $n$, we can associate a number (real or complex) called determinant of $A.$

II. Determinant is a function which associates each square matrix with a unique number (real or complex).