Equality of Matrices

Q. Let $A$ and $B$ be two $3\times 3$ real matrices such that $(A^2–B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3{B}^2=A^2{B}^3$, then the value of the determinant of the matrix $A^3+B^3$ is equal to

1. $1$
2. $2$
3. $4$
4. $0$

Q. Let $X=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ and $A=\left[\begin{array}{ccc}-1& 2& 3\\ 0& 1& 6\\ 0& 0& -1\end{array}\right]$. For $k\in N$, if $X^{\prime }{A}^{k}X=33$, then $k$ is equal to

Q. Let $A=\left[\begin{array}{cc}1& 2\\ -1& 4\end{array}\right].$ If $A^{-1}=\alpha I+\beta A,\alpha ,\beta \in R,I$ is $2\times 2$ identity matrix, then $4(\alpha -\beta )$ is

1. $\frac{8}{3}$
2. $5$
3. $4$
4. $2$

Q.

If $\alpha$ and $\beta$are the roots of the equation$x^2-x+1=0$, then${\alpha }^{2009}+{\beta }^{2009}$is equal to

1. $-2$

2. $-1$

3. $1$

4. $2$

Q. If $x\left[\begin{array}{c}2\\ 3\end{array}\right]+y\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$, find the values of $x$ and $y$

Q. If an idempotent matrix is also skew symmetrix, then it must be

1. an involuntary matrix
2. an identity matrix
3. an orthogonal matrix
4. a null matrix

Q. Given , find the values of x , y , z and w .

Q.

If $A=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right],$ Then the value of $\alpha$ for which $A^2=B$ is

1. 1

2. -1

3. $i$

4. no real values of $\alpha$

Q.

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

1. $nA-(n-1)I$

2. $nA-I$

3. $2^{n+1}A-(n-1)I$

4. $2^{n-1}A-I$

Q. If $A=\left[\begin{array}{cc}0& x\\ y& 0\end{array}\right]$ and $A^3+A=O,$ then which of the following is correct

1. $xy=-1$
2. $xy=0$
3. $xy=\frac{1}{2}$
4. $xy=1$

Q. Let $\Delta \left(x\right)=\left|\begin{array}{ccc}2x^3-3x^2& 5x+7& 2\\ 4x^3-7x& 3x+2& 1\\ 7x^3-8x^2& x-1& 3\end{array}\right|=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4$

Column - I	Column - II
(A) $a_0$ equals to	(p) -73
(B) $a_1$ 3quals to	(q) 46
(C) $a_0$ equals to	(r) -43
(D) $a_4$ equals to	(s) 161
	(t) 0

1. A(t), B(s), C(r), D(p)
2. A(t), B(s), C(p), D(r)

Q. If $A=\left[\begin{array}{cc}0& \alpha \\ 0& 0\end{array}\right]$ and $(A+I{)}^{50}-50A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$ then the value of $a+b+c+d$ is

Q. Matrices $A$ and $B$ satisfy $AB=B^{-1}$, where $B=\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right].$ Then the value of the scalar $k$ for which $kA-2B^{-1}+I=O$ is

Q. Find the number of different matrices that can be formed with elements 0, 1, 2 or 3. Each matrix having 4 elements.

Q. If $A=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$, then the value of $\alpha$ for which $A^2=B$ is:

1. $1$
2. $2$
3. $4$
4. No real values.

Q. Consider the following statement :
$A:\text{If}AB=A\text{and}BA=B\text{then}{A}^n+B^n=A+B$
$R:A\text{and}B$ are idempotent matrices.
Then the correct option is

1. Both statements $A$ and $R$ are True and $R$ is correct explanation of $A.$
2. Both $A$ and $R$ are True but $R$ is not correct explanation to $A$
3. $A$ is true $R$ is False
4. $A$ is false $R$ is true

Q. Let $A$ and $B$ be two $n\times n$ matrices such that $det\left(A\right)\ne 0,A+B=(AB{)}^2$ and $BAB=A+I$, then
$\left(I\right)A^{-1}=(A^4-I)\left(II\right)B^5-A^5=A+B\left(III\right)A^9=A^4+A+I\left(IV\right)A^2{B}^2=B{A}^{2}B$

Which of the following is correct?
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. Only $\left(I\right)$ is true
2. Only $\left(I\right)$ and $\left(IV\right)$ are true
3. Only $\left(I\right),\left(III\right)$ and $\left(IV\right)$ are true
4. All are true

Q. The number of value(s) of $x$ satisfying the equation $x^2-2x=1+\sqrt{1+x},x\in (2,\infty )$ is

1. $0$
2. $3$
3. $1$
4. $2$

Q. Evaluate the determinant $\left|\begin{array}{cc}2& 4\\ -5& -1\end{array}\right|$

Q. The determinant $\left|\begin{array}{ccc}\sin A& \cos A& \sin A+\cos B\\ \sin B& \cos A& \sin B+\cos B\\ \sin C& \cos A& \sin C+\cos B\end{array}\right|$ is equal to

1. $0$
2. $1$
3. ${\sin }^{2}A+\cos A$
4. $\sin A+\cos B+\cos A$

Q. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],$ then $A^4-5A^3-A^2-4A-2I=$

1. $0$
2. $A$
3. $I$
4. 
   $2I$
   

Q. $f\left(x\right)=\left[\frac{x^2+1}{x^2+\left[|x|\right]+1}\right]$ is discontinuous at k points, then k is -
(where [.] denotes greatest integer function)

Q. The value of $\theta$ satisfying $\left|\begin{array}{ccc}1& 1& \sin 3\theta \\ -4& 3& \cos 2\theta \\ 7& -7& -2\end{array}\right|=0$ in $[0,2\pi ]$ is

Q. Find x and y , if

Q. 6. Find the values of x, y and z from the following equations:rtyz 9| X+2 |=152x+y 21「6 215+z xy| |5825(iii)x 51 5

Q. The value of determinant $\Delta =\left|\begin{array}{ccc}{x}^2+4& 2x& 1-x\\ 5x+1& 2x& 1\\ x+2& x& 2\end{array}\right|,(x\ne 0)$ is equal to

1. $12x^2-9x$
2. $12x^2+12x$
3. $-12x^2+9x$
4. $-12x^2-9x$

Q. A square matrix $A$ is said to be orthogonal if $A^{\prime }A=A{A}^{\prime }=I_n,A^{\prime }$ is transpose of $A.$
If $A$ and $B$ are orthogonal matrices, of the same order, then which one of the following is an orthogonal matrix

1. $AB$
2. $A+B$
3. $A+iB$
4. $i(A+B)$

Q.

If matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ where a, b, c are real positive numbers, abc = 1 and $A^{T}A=I,$ then the value of $a^3+b^3+c^3$ is ___

Q. Matrices $A$ and $B$ will be inverse of each other only if

1. $AB=BA=I$
2. $AB=0,BA=I$
3. $AB=BA=0$
4. $AB=BA$

Q.

If $\alpha$and $\beta$are the roots of the quadratic equation$x^2+px+q=0$, then the values of ${\alpha }^3+{\beta }^3$ and ${\alpha }^4+{\alpha }^2{\beta }^2+{\beta }^4$ are respectively

1. $3pq–p^3$and $p^4–3p^{2}q+3q^2$

2. $-p(3q–p^2)$and $(p^2–q)(p^2+3q)$

3. $pq–4$and $p^4–q^4$

4. $3pq–p^3$ and $(p^2–q)(p^2–3q)$