Properties of Matrix Multiplication

Q. If $[A{]}_{m\times n}$ and $[B{]}_{n\times p}$ are two matrices, then the order of AB will be .

1. $n\times n$
2. $m\times p$
3. $p\times m$

Q.

If $[A{]}_{m\times n}$ and $[B{]}_{n\times p}$ are two matrices, then the order of AB will be n $\times$n.

1. True

2. False

Q. If you multiply a $2\times 2$ matrix and a $2\times 1$ matrix the product is a $2\times 1$ matrix?

1. True
2. False

Q. If three matrices A, B and C follows Right Distributive Property, then their orders respectively can be:

1. $p\times q,q\times randq\times r$
2. $p\times q,p\times qandq\times r$
3. $p\times q,q\times randr\times s$
4. $p\times q,p\times qandp\times q$

Q. Let $A=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$. The only correct statement about the matrix A is

1. A is a zero matrix
2. A = (-1)I, where I is a unit matrix.
3. $A^2=I$
4. $A^{-1}$ does not exist.

Q. $\text{If}A=\left[\begin{array}{cc}5& 0\\ 2& 2\end{array}\right],\text{then find}A\times I_{2\times 2}.$

1. $$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$$
2. $$\left[\begin{array}{cc}5& 0\\ 2& 2\end{array}\right]$$
3. $$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
4. $$\left[\begin{array}{cc}-5& 0\\ -2& -2\end{array}\right]$$

Q. If $A$ and $B$ are any two matrices, then

1. $AB$ may or may not be defined
2. $AB=BA$
3. $AB=I$
4. $AB=0$

Q. Given $A=\left[\begin{array}{cc}2& 1\\ 3& 0\end{array}\right],B=\left[\begin{array}{cc}1& 1\\ 5& 2\end{array}\right]andC=\left[\begin{array}{cc}-3& -1\\ 0& 0\end{array}\right]$
$2A-3B+C$

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.

Matrix A has x rows and x + 5 columns. Matrix B has y rows and 11 - y columns. Both AB and BA exist.Then, the value of x and y is

1. 3, 8

2. 4, 8

3. 3, 9

4. 8, 3

Q.

If $A^T=\left[\begin{array}{cc}3& 4\\ -1& 2\\ 0& 1\end{array}\right]\text{and}B=\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]$,

then find $A^T-B^T$

Q. Let$A=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 2\\ 7& 4\end{array}\right]$ and $C=\left[\begin{array}{cc}2& 5\\ 3& 8\end{array}\right]$. Find a matrix D such that $CD-AB=0$
$\left[5\right]$

Q. (a) $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 7\\ 3& 6& 10\end{array}\right]$
(b) $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 4& 10& 18\end{array}\right]$

Q. $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, if $ad-bc=0$, $A^2=A$.Find $a+d$.

Q.

If A and B are two matrices such that AB=B and BA=A, then $A^2+B^2$=

1. $2AB$

2. $A+B$

3. $AB$

4. $2BA$

Q.

If $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]$ and $A^2$ = I then x = ___

Q. Let,
$A=\left[\begin{array}{ccc}4& 6& -1\\ 3& 0& 2\\ 1& -2& 5\end{array}\right],B=\left[\begin{array}{cc}2& 4\\ 0& 1\\ -1& 2\end{array}\right]$
$\text{and}C=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$
The expression which is not defined is:

1. $B^{\prime }B$
2. $CAB$
3. $A+B^{\prime }$
4. $A^2+A$

Q.

Order of matrix A is 3 $\times$ 5 and that of B is 2 $\times$ 3. Then then order of the matrix BA is

1. 2 $\times$ 3

2. 3 $\times$ 2

3. 2 $\times$ 5

4. 5 $\times$ 2

Q. Let $A=\left[\begin{array}{cc}a& b\\ 0& 1\end{array}\right]$ where $a\ne 0$. Then $A^n=$?

1. $A^n=\left[\begin{array}{cc}{a}^n& \frac{b(a^n-1)}{(a-1)}\\ 0& 1\end{array}\right]$
2. $A^n=\left[\begin{array}{cc}\frac{a(b^n-1)}{(b-1)}& \frac{b(a^n-1)}{(a-1)}\\ 0& 1\end{array}\right]$
3. $A^n=\left[\begin{array}{cc}{a}^n& b^n\\ 0& 1\end{array}\right]$
4. none of these

Q. Find matrices $X$ and $Y$, if $2X-Y=\left[\begin{array}{ccc}6& -6& 0\\ -4& 2& 1\end{array}\right]$ and $X+2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]$.

Q. Prove that $\left|\begin{array}{ccc}1& a& bc\\ 1& b& ca\\ 1& c& ab\end{array}\right|=(a-b)(b-c)(c-a)$.

Q. Find the area of the quadrilateral whose vertices are $A(1,1),B(7,-3),C(12,2)$ and $D(7,21)$

1. $123$ square units
2. $132$ square units
3. $142$ square units
4. $256$ square units