Nilpotent Matrix

Q. If A and B are two matrices such that AB = B and BA = A, then

1. $(A^6-B^5{)}^3=A-B$
2. $A-Bisidempotent$
3. $A-Bisnilpotent$
4. $(A^5-B^5{)}^3=A^3-B^3$

Q. If the orthogonal square matrix A and B satisfy, det A + det B = 0, then the value of det(A+B) =

1. Equals 0
2. Equals 1
3. Equals 1 or -1, and both these values are of possible single choice
4. Equals -1

Q. $\left(i\right)$ If $A=\left[\begin{array}{cccc}3& \sqrt{3}& 2& \\ 4& 2& 0& \end{array}\right]$ and $B=\left[\begin{array}{ccc}2& -1& 2\\ 1& 2& 4\end{array}\right]$, verify that $(A^{\prime }{)}^{\prime }=A.$
$\left(ii\right)$ If $A=\left[\begin{array}{cccc}3& \sqrt{3}& 2& \\ 4& 2& 0& \end{array}\right]$ and $B=\left[\begin{array}{ccc}2& -1& 2\\ 1& 2& 4\end{array}\right]$, verify that $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }.$
$\left(iii\right)$ If $A=\left[\begin{array}{cccc}3& \sqrt{3}& 2& \\ 4& 2& 0& \end{array}\right]$ and $B=\left[\begin{array}{ccc}2& -1& 2\\ 1& 2& 4\end{array}\right]$, verify that $(kB{)}^{\prime }=(k{B}^{\prime })$ where $k$ is any constant.

Q. Let $A$ and $B$ be non singular square matrices of order $3$ satisfying $A+adjA=B$ and $|A|=3,$ then $|\left(adjA\right)B-3I|$ is equal to:

1. $9$
2. $27$
3. $81$
4. $243$

Q. A square matrix $A=[a_{ij}{]}_{n\times n}$, if $a_{ij}=0$ for $i>j,$ then that matrix is known as

1. Upper triangular matrix
2. Lower triangluar matrix
3. Unit matrix
4. Null matrix

Q. The nilpotency index of matrix $\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ -1& -2& -3\end{array}\right]$ is

Q. If $A$ and $B$ are two matrices such that $AB=A$ and $BA=B,$ then

1. $(A^3-B^3{)}^5=(A^5-B^5{)}^3$
2. $(A^4-B^4{)}^3=A^{12}-B^{12}$
3. $(A-B)$ is idempotent
4. $(A^{10}-B^{10})$ is nilpotent

Q. If A and B are two matrices such that AB = B and BA = A, then

1. $A-Bisidempotent$
2. $(A^5-B^5{)}^3=A^3-B^3$
3. $A-Bisnilpotent$
4. $(A^6-B^5{)}^3=A-B$

Q. The value of $a$ if $\left[\begin{array}{ccc}1& a& 1\end{array}\right]\left[\begin{array}{c}2a\\ a\\ 2\end{array}\right]=\left[\begin{array}{c}1\end{array}\right]$

1. $1$
2. $-1$
3. $2$
4. $-2$

Q. We call a a good number if inequality $\frac{{2x}^2+2x+3}{x^2+x+1}\le a$ is satisfied for any real x. Find the smallest good integral number

Q.

$A=\left[\begin{array}{cc}2& 4\\ -1& -2\end{array}\right]$ is a nilpotent matrix.

1. True

2. False

Q. If A = $\left[\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right]$ then det (Adj A) =

1. 4
2. 16
3. 8
4. 2

Q. $A$ and $B$ are two non-zero square matrices such that $AB=0$. Then

1. Both $A$ and $B$ are singular
2. Either of them is singular
3. Neither matrix is singular
4. None of these

Q.

A square matrix A is called Nilpotent if there exists a positive integer m such that $A^m$ = I.

1. True

2. False

Q. If $a<b<c<d$ & $xϵR$ then the least value of the function,
$f\left(x\right)=|x-a|+|x-b|+|x-c|+|x-d|$ is

1. $c+d-b-a$
2. $c+d-b+a$
3. $c-d+b-a$
4. $c-d+b+a$

Q. The difference between the greatest and least values of the function $F\left(x\right)={\int }_0^x(t+1)dt$ on $[2,3]$ is

1. $3$
2. $2$
3. $7/2$
4. $11/2$

Q. lf the adjoint of a $3\times 3$ matrix $P$ is $\left[\begin{array}{ccc}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right]$, then the possible value(s) of the determinant of $P$ is (are):

1. $1$
2. $-1$
3. $2$
4. $-2$

Q. The value of the determinant $\left|\begin{array}{ccc}1+i& 1-i& i\\ 1+i& i& 1+i\\ i& 1+i& 1-i\end{array}\right|$

1. $2+5i$
2. $7-4i$
3. $4+7i$
4. $-2+5i$

Q.

The index of the matrix A -
Where $A=\left[\begin{array}{cc}2& 4\\ -1& -2\end{array}\right]$___

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

Q.

$A=\left[\begin{array}{cc}2& 4\\ -1& -2\end{array}\right]$ is a nilpotent matrix.

1. True

2. False

Q. $A=\left[\begin{array}{ccc}2& 3& 1\\ 4& 1& 5\\ 3& 9& 7\end{array}\right]$. Then the additive inverse of $A$ is

1. $\left[\begin{array}{ccc}-2& -3& 1\\ 4& -1& -5\\ -3& 9& -7\end{array}\right]$
2. $\left[\begin{array}{ccc}-2& -3& -1\\ -4& -1& -5\\ -3& -9& -7\end{array}\right]$
3. $\left[\begin{array}{ccc}2& -3& -1\\ -4& 1& -5\\ -3& -9& 7\end{array}\right]$
4. $\left[\begin{array}{ccc}-2& -3& -1\\ -4& -1& -5\\ -3& 9& -7\end{array}\right]$

Q. Given $A=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$.$I$ is a unit matrix of order $2$. Find all possible matrix $X$ such that $AX=I$

1. $X$ does not exist
2. $X$ is a unit matrix
3. $X$ is a identity matrix
4. None of these