Square Matrix

Q. $A$ and $B$ are two square matrices such that $A^{2}B=BA$ and if $(AB{)}^{10}=A^k.B^{10}$ then the value of $k-1020$ is

Q. 14. What is nilpotent matrix?

Q.

Let A be the non-singular square matrix of order $3\times 3$ then |adj A| is equal to

a) $|A|$
b) $|A{|}^2$
c) $|A{|}^3$
d) $3|A|$

Q. Find $AB,$ if $A=\left[\begin{array}{cc}0& -1\\ 0& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}3& 5\\ 0& 0\end{array}\right]$

Q. Let matrices $A=\left[\begin{array}{ccc}2x+1& 3y& \\ 0& y^2-5y& \end{array}\right],B=\left[\begin{array}{ccc}x+3& y^2+2& \\ 0& -6& \end{array}\right]$ are equal, then which of the following is(are) not correct

1. $x=2,y=-2$
2. $x=-2,y=2$
3. $x=1,y=1$
4. $x=2,y=2$

Q. 2 What are involutary matrix

Q. If two matrices $A$ and $B$ are such that they follow commutative property, then $A^4{B}^2$ is equal to

1. $B^2{A}^4$
2. $B{A}^{3}B$
3. $A^3{B}^{2}A$
4. $A{B}^{3}A$

Q. Compute the indicated product
$\left(i\right)$ $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$
$\left(ii\right)$ $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[\begin{array}{ccc}2& 3& 4\end{array}\right]$
$\left(iii\right)$ $\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$
$\left(iv\right)$ $\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\\ 4& 5& 6\end{array}\right]\left[\begin{array}{ccc}1& -3& 5\\ 0& 2& 4\\ 3& 0& 5\end{array}\right]$
$\left(v\right)$ $\left[\begin{array}{cc}2& 1\\ 3& 2\\ -1& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 1\\ -1& 2& 1\end{array}\right]$
$\left(vi\right)$ $\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]$

Q. Construct a $3\times 4$ matrix, whose elements are given by
(i) $a_{ij}=\frac{1}{2}|-3i+j|$
(ii) $a_{ij}=2i-j$

Q. The rank of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 3& 6& 9\\ 1& 2& 3\end{array}\right]$, is

1. 1
2. 2
3. 3
4. none of these

Q.

Show that:
$\left(i\right)$ $\left[\begin{array}{cc}5& -1\\ 6& 7\end{array}\right]\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]\ne \left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]\left[\begin{array}{cc}5& -1\\ 6& 7\end{array}\right]$

$\left(ii\right)$ $\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\ne \left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$

Q. If $A=\left[\begin{array}{c}-2\\ 4\\ 5\end{array}\right],B=\left[\begin{array}{ccc}1& 3& -6\end{array}\right]$, verify that $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }.$

Q.

How do you write interval notation $?$

Q.

Choose the correct answer in questions

Let A be a square matrix of order $3\times 3$, then $|kA|$ is equal
a) $k|A|$
b) $k^2|A|$
c) $k^3|A|$
d) $3k|A|$

Q. If $A$ is a square matrix such that $A^2+A+2I=O,$ and $I$ is an identity matrix of same order as that of $A$, then which of the following is/are true:

1. $A$ is non-singular
2. $A$ is symmetric
3. $A$ can't be skew-symmetric
4. $A^{-1}=-\frac{1}{2}(A+I)$

Q. The rank of a null matrix is

1. 0
2. 1
3. does not exist
4. none of these

Q. If A, B are two square matrices such that $AB=A$ and $BA=B$, then prove that $B^2=B$

Q. $A=[a_{ij}{]}_{m\times n}$ is a square matrix, if

1. $m=n$
2. None of these
3. $m<n$
4. $m>n$

Q. A matrix whose number of rows is equal to the number of its columns is called a ________.

1. row matrix
2. square matrix
3. null matrix
4. none of the

Q. If $A$ is a non-singular square matrix of order $3\times 3,$ find $|adjA|$

Q.

Evaluate the determinants.

$\left[\begin{array}{cc}2& 4\\ -5& -1\end{array}\right]$

Q. Let $A,B$ and $C$ be square matrices of order $3\times 3.$ If $A$ is invertible and $(A-B)C=B{A}^{-1},$ then

1. $C(A-B)=A^{-1}B$
2. $C(A-B)=B{A}^{-1}$
3. $(A-B)C=A^{-1}B$
4. All of the above

Q. 8.A = [aij, xn is a square matrix, if(A) m<1n(B) m>in(C) m=n(D) None of these

Q.

A square matrix is having p number of rows. What is ‘j’ if $a_{ij}$ is situated in the last row and second last column

1. 2

2. p + 2

3. p - 1

4. 2p

Q. Let $A$ and $B$ be two square matrices of order $3$ and $AB=O_3$, where $O_3$ denotes the null matrix of order $3$. Then,

1. must be $A=O_3,B=O_3$
2. if $A\ne O_3$, must be $B\ne O_3$
3. if $A=O_3$, must be $B\ne O_3$
4. may be $A\ne O_3,B\ne O_3$

Q. If a matrix has $13$ elements, then the possible dimensions (orders) of the matrix are

1. $1\times 13$ or $13\times 1$
2. $1\times 26$ or $26\times 1$
3. $2\times 13$ or $13\times 2$
4. $13\times 13$

Q. Find the values of $a,b,c,d$
$\left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right]$

Q. If $\left[\begin{array}{cc}a+2b& 2c-d\\ c+4d& 4b-a\end{array}\right]=\left[\begin{array}{cc}10& 2\\ 37& 14\end{array}\right]$, then the values of $a,b,c,d$ respectively is

$\left(a\right)2,4,6,8\left(b\right)2,4,5,8$
$\left(c\right)2,4,5,7\left(d\right)1,4,5,8$

Q. Following questions contains statements given in two columns, which have to be matched. The statements in $Column-I$ are labelled as A, B, C and D while the statements in $Column-II$ are labelled as p, q, r and s. Any given statement in $Column-I$ can have correct matching with $ONE$ statement in $Column-II$.

Q. The matrix $P=\left[\begin{array}{ccc}0& 0& 4\\ 0& 4& 0\\ 4& 0& 0\end{array}\right]$ is a

1. square matrix
2. unit matrix
3. diagonal matrix
4. none of these