If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

In the matrix $A=\left[\begin{array}{ccc}a & 1 & x \\ 2 & \sqrt{3} & x^2-y \\ 0 & 5 & \frac{-2}{5}\end{array}\right]$, write

(i) the order of the matrix $A$.

(ii) the number of elements.

(iii) elements $a_{23}, a_{31}$ and $a_{12}$.

Construct $a_{2 \times 2}$ matrix, where

(i) $a_{i j}=\frac{(i-2 j)^2}{2}\quad$ (ii) $a_{i j}=|-2 i+3 j|$

Construct a $3 \times 2$ matrix whose elements are given by $a_{i j}=e^{i \cdot x}=\sin j x$.

Find the values of $a$ and $b$, if $A=B$, where $$ A=\left[\begin{array}{cc} a+4 & 3 b \\ 8 & -6 \end{array}\right] \text { and } B=\left[\begin{array}{cc} 2 a+2 & b^2+2 \\ 8 & b^2-5 b \end{array}\right]$$

If possible, find the sum of the matrices $A$ and $B$, where $A=\left[\begin{array}{cc}\sqrt{3} & 1 \\ 2 & 3\end{array}\right]$ and $B=\left[\begin{array}{lll}x & y & z \\ a & b & c\end{array}\right]$.

If $X=\left[\begin{array}{rrr}3 & 1 & -1 \\ 5 & -2 & -3\end{array}\right]$ and $Y=\left[\begin{array}{rrr}2 & 1 & -1 \\ 7 & 2 & 4\end{array}\right]$, then find

(i) $X+Y$.

(ii) $2 X-3 Y$.

(iii) a matrix $Z$ such that $X+Y+Z$ is a zero matrix.

Find non-zero values of $x$ satisfying the matrix equation $$ x\left[\begin{array}{cc} 2 x & 2 \\ 3 & x \end{array}\right]+2\left[\begin{array}{ll} 8 & 5 x \\ 4 & 4 x \end{array}\right]=2\left[\begin{array}{cc} \left(x^2+8\right) & 24 \\ (10) & 6 x \end{array}\right]$$

If $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, then show that $$(A+B)(A-B) \neq A^2-B^2 .$$

Find the value of $x$, if $\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=0$.

Show that $A=\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right]$ satisfies the equation $A^2-3 A-7 I=0$ and hence find the value of $A^{-1}$.

Find the matrix $A$ satisfying the matrix equation $$ \left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] A\left[\begin{array}{cc} -3 & 2 \\ 5 & -3 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] . $$

Find $A$, if $\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right] A=\left[\begin{array}{lll}-4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3\end{array}\right]$.

If $A\left[\begin{array}{cc}3 & -4 \\ 1 & 1 \\ 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{lll}2 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$, then verify $(B A)^2 \neq B^2 A^2$.

If possible, find the value of $B A$ and $A B$, where $$ A=\left[\begin{array}{lll} 2 & 1 & 2 \\ 1 & 2 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 4 & 1 \\ 2 & 3 \\ 1 & 2 \end{array}\right] \text {. }$$

Show by an example that for $A \neq 0, B \neq 0$ and $A B=0$.

Given, $A=\left[\begin{array}{lll}2 & 4 & 0 \\ 3 & 9 & 6\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 4 \\ 2 & 8 \\ 1 & 3\end{array}\right]$. is $(A B)^{\prime}=B^{\prime} A^{\prime}$ ?

Solve for $x$ and $y, x\left[\begin{array}{l}2 \\ 1\end{array}\right]+y\left[\begin{array}{l}3 \\ 5\end{array}\right]+\left[\begin{array}{c}-8 \\ -11\end{array}\right]=0$.

If $X$ and $Y$ are $2 \times 2$ matrices, then solve the following matrix equations for $X$ and $Y$

$$2 X+3 Y=\left[\begin{array}{ll} 2 & 3 \\ 4 & 0 \end{array}\right], 3 X+2 Y=\left[\begin{array}{cr} -2 & 2 \\ 1 & -5 \end{array}\right]$$

If $A=\left[\begin{array}{ll}3 & 5\end{array}\right]$ and $B=\left[\begin{array}{ll}7 & 3\end{array}\right]$, then find a non-zero matrix $C$ such that $A C=B C$.

Give an example of matrices $A, B$ and $C$, such that $A B=A C$, where $A$ is non-zero matrix but $B \neq C$.

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

(i) $(A B) C=A(B C)$.

(ii) $A(B+C)=A B+A C$.

If $P=\left[\begin{array}{ccc}x & 0 & 0 \\ 0 & \mathrm{y} & 0 \\ 0 & 0 & \mathrm{z}\end{array}\right]$ and $Q=\left[\begin{array}{ccc}\mathrm{a} & 0 & 0 \\ 0 & \mathrm{~b} & 0 \\ 0 & 0 & \mathrm{c}\end{array}\right]$, then prove that $P Q=\left[\begin{array}{ccc}x \mathrm{a} & 0 & 0 \\ 0 & \mathrm{yb} & 0 \\ 0 & 0 & \mathrm{zc}\end{array}\right]=Q P$

If $\left[\begin{array}{lll}2 & 1 & 3\end{array}\right]\left[\begin{array}{ccc}-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right]\left[\begin{array}{c}1 \\ 0 \\ -1\end{array}\right]=A$, then find the value of $A$.

If $A=[21], B=\left[\begin{array}{lll}5 & 3 & 4 \\ 8 & 7 & 6\end{array}\right]$ and $C=\left[\begin{array}{ccc}-1 & 2 & 1 \\ 1 & 0 & 2\end{array}\right]$, then verify that $A(B+C)=(A B+A C)$.

If $A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{array}\right]$, then verify that $A^2+A=(A+I)$, where $I$ is $3 \times 3$ unit matrix.

If $A=\left[\begin{array}{ccc}0 & -1 & 2 \\ 4 & 3 & -4\end{array}\right]$ and $B=\left[\begin{array}{ll}4 & 0 \\ 1 & 3 \\ 2 & 6\end{array}\right]$, then verify that

(i) $\left(A^{\prime}\right)^{\prime}=A$

(ii) $(A B)^{\prime}=B^{\prime} A^{\prime}$

(iii) $(k A)^{\prime}=\left(k A^{\prime}\right)$.

If $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 1 \\ 5 & 6\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 2 \\ 6 & 4 \\ 7 & 3\end{array}\right]$, then verify that

(ii) $(A-B)^{\prime}=A^{\prime}-B^{\prime}$.

Show that $A^{\prime} A$ and $A A^{\prime}$ are both symmetric matrices for any matrix A.

Let $A$ and $B$ be square matrices of the order $3 \times 3$. Is $(A B)^2=A^2 B^2$ ? Give reasons.

Show that, if $A$ and $B$ are square matrices such that $A B=B A$, then $(A+B)^2=A^2+2 A B+B^2$

If $A=\left[\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right], B=\left[\begin{array}{cc}4 & 0 \\ 1 & 5\end{array}\right], C=\left[\begin{array}{cc}2 & 0 \\ 1 & -2\end{array}\right], a=4$, and $b=-2$, then show that

(i) $A+(B+C)=(A+B)+C$

(ii) $A(B C)=(A B) C$

(iii) $(a+b) B=a B+b B$

(iv) $a(C-A)=a C-a A$

(v) $\left(A^T\right)^T=A$

(vi) $(b A)^T=b A^T$

(vii) $(A B)^T=B^T A^T$

(viii) $(A-B) C=A C-B C$

(ix) $(A-B)^T=A^T-B^T$

If $A=\left[\begin{array}{cc}\cos q & \sin q \\ -\sin q & \cos q\end{array}\right]$, then show that $A^2=\left[\begin{array}{cc}\cos 2 q & \sin 2 q \\ -\sin 2 q & \cos 2 q\end{array}\right]$.

If $A=\left[\begin{array}{cc}0 & -x \\ x & 0\end{array}\right], B=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ and $x^2=-1$, then show that $(A+B)^2=A^2+B^2$.

Verify that $A^2=I$, when $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4\end{array}\right]$.

Prove by mathematical induction that $\left(A^{\prime}\right)^n=\left(A^n\right)^{\prime}$ where $n \in N$ for any square matrix $A$.

Find inverse, by elementary row operations (if possible), of the following matrices. (i) $\left[\begin{array}{cc}1 & 3 \\ -5 & 7\end{array}\right]$ (ii) $\left[\begin{array}{cc}1 & -3 \\ -2 & 6\end{array}\right]$

If $\left[\begin{array}{cc}x y & 4 \\ z+6 & x+y\end{array}\right]=\left[\begin{array}{cc}8 & w \\ 0 & 6\end{array}\right]$, then find the values of $x, y, z$ and $w$.

If $A=\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]$ and $B=\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right]$, then find a matrix $C$ such that $3 A+5 B+2 C$ is a null matrix.

If $A=\left[\begin{array}{cc}3 & -5 \\ -4 & 2\end{array}\right]$, then find $A^2-5 A-14 I$. Hence, obtain $A^3$.

Find the values of $a, b, c$ and $d$, if $$ 3\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{cc} a & 6 \\ -1 & 2 \end{array}\right]+\left[\begin{array}{cc} 4 & a+b \\ c+d & 3 \end{array}\right] z . $$

Find the matrix $A$ such that $\left[\begin{array}{cc}2 & -1 \\ 1 & 0 \\ -3 & 4\end{array}\right] A=\left[\begin{array}{ccc}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]$

If $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 1\end{array}\right]$, then find $A^2+2 A+7 I$

If $A=\left[\begin{array}{rr}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ and $A^{-1}=A^{\prime}$, then find the value of $\alpha$.

If matrix $\left[\begin{array}{ccc}0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0\end{array}\right]$ is a skew-symmetric matrix, then find the values of $a, b$ and $c$.

If $\quad P(x)=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]$, then show that $P(x) \cdot P(y)=P(x+y)$ $=P(y) \cdot P(x)$.

If $A$ is square matrix such that $A^2=A$, then show that $(I+A)^3=7 A+I$.

If $A, B$ are square matrices of same order and $B$ is a skew-symmetric matrix, then show that $A^{\prime} B A$ is skew-symmetric.

If $A B=B A$ for any two square matrices, then prove by mathematical induction that $(A B)^n=A^n B^n$.

Find $x, y$ and $z$, if $A=\left[\begin{array}{ccc}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]$ satisfies $A^{\prime}=A^{-1}$.

Q. 51 If possible, using elementary row transformations, find the inverse of the following matrices. (i) $\left[\begin{array}{ccc}2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3\end{array}\right]$ (ii) $\left[\begin{array}{ccc}2 & 3 & -3 \\ -1 & -2 & 2 \\ 1 & 1 & -1\end{array}\right]$ (iii) $\left[\begin{array}{ccc}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$

2 Express the matrix $\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.

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

Total number of possible matrices of order $3\times3$ with each entry 2 or 0 is

$\left[\begin{array}{ll}2 x+y & 4 x \\ 5 x-7 & 4 x\end{array}\right]=\left[\begin{array}{cc}7 & 7 y-13 \\ y & x+6\end{array}\right]$, then the value of $x+y$ is

If $A=\frac{1}{\pi}\left[\begin{array}{cc}\sin ^{-1}(x \pi) & \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \sin ^{-1}\left(\frac{x}{\pi}\right) & \cot ^{-1}(\pi x)\end{array}\right]$ and $B=\frac{1}{\pi}\left[\begin{array}{cc}-\cos ^{-1}(x \pi) & \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \sin ^{-1}\left(\frac{x}{\pi}\right) & -\tan ^{-1}(\pi x)\end{array}\right]$, then $A-B$ is equal to

If $A$ and $B$ are two matrices of the order $3 \times \mathrm{m}$ and $3 \times n$, respectively and $m=n$, then order of matrix $(5 A-2 B)$ is

If $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$, then $A^2$ is equal to

If matrix $A=\left[a_{i j}\right]_{2 \times 2}$, where $a_{i j}=1$, if $i \neq j=0$ and if $i=j$, then $A^2$ is equal to

The matrix $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{array}\right]$ is a

The matrix $\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]$ is a

If $A$ is matrix of order $m \times n$ and $B$ is a matrix such that $A B^{\prime}$ and $B^{\prime} A$ are both defined, then order of matrix $B$ is

If $A$ and $B$ are matrices of same order, then $\left(A B^{\prime}-B A^{\prime}\right)$ is

If $A$ is a square matrix such that $A^2=I$, then $(A-I)^3+(A+I)^3-7 A$ is equal to

For any two matrices $A$ and $B$, we have

Q. 66 On using elementary column operations $C_2 \rightarrow C_2-2 C_1$ in the following matrix equation $\left[\begin{array}{cc}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]$, we have

On using elementary row operation $R_1 \rightarrow R_1-3 R_2$ in the following matrix equation $\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$, we have

........... matrix is both symmetric and skew-symmetric matrix.

Sum of two skew-symmetric matrices is always ........... matrix.

The negative of a matrix is obtained by multiplying it by ............ .

The product of any matrix by the scalar ............. is the null matrix.

A matrix which is not a square matrix is called a ........... matrix.

Matrix multiplication is ........... over addition.

If $A$ is a symmetric matrix, then $A^3$ is a ............ matrix.

If $A$ is a skew-symmetric matrix, then $A^2$ is a .............. .

If $A$ and $B$ are square matrices of the same order, then

(i) $(A B)^{\prime}=$ ...........

(ii) $(k A)^{\prime}=$ ........... (where, $k$ is any scalar)

(iii) $[k(A-B)]^{\prime}=$ ..............

If $A$ is a skew-symmetric, then $k A$ is a ............ (where, $k$ is any scalar).

If $A$ and $B$ are symmetric matrices, then

(i) $A B-B A$ is a .............

(ii) $B A-2 A B$ is a ...........

If $A$ is symmetric matrix, then $B^{\prime} A B$ is ............ .

If $A$ and $B$ are symmetric matrices of same order, then $A B$ is symmetric if and only if ............. .

In applying one or more row operations while finding $A^{-1}$ by elementary row operations, we obtain all zeroes in one or more, then $A^{-1} \ldots \ldots .$.

A matrix denotes a number.

Matrices of any order can be added.

Two matrices are equal, if they have same number of rows and same number of columns.

Matrices of different order cannot be subtracted.

Matrix addition is associative as well as commutative.

Matrix multiplication is commutative.

A square matrix where every element is unity is called an identity matrix.

If $A$ and $B$ are two square matrices of the same order, then $A+B=B+A$.

If $A$ and $B$ are two matrices of the same order, then $A-B=B-A$.

If matrix $A B=0$, then $A=0$ or $B=0$ or both $A$ and $B$ are null matrices.

Transpose of a column matrix is a column matrix.

If $A$ and $B$ are two square matrices of the same order, then $A B=B A$.

If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

If $A$ and $B$ are any two matrices of the same order, then $(A B)^{\prime}=A^{\prime} B^{\prime}$.

Q. 96 If $(A B)^{\prime}=B^{\prime} A^{\prime}$, where $A$ and $B$ are not square matrices, then number of rows in $A$ is equal to number of columns in $B$ and number of columns in $A$ is equal to number of rows in $B$.

If $A, B$ and $C$ are square matrices of same order, then $A B=A C$ always implies that $B=C$.

$A A^{\prime}$ is always a symmetric matrix for any matrix $A$.

If $A=\left|\begin{array}{ccc}2 & 3 & -1 \\ 1 & 4 & 2\end{array}\right|$ and $B=\left|\begin{array}{cc}2 & 3 \\ 4 & 5 \\ 2 & 1\end{array}\right|$, then $A B$ and $B A$ are defined and equal.

If $A$ is skew-symmetric matrix, then $A^2$ is a symmetric matrix.

$(A B)^{-1}=A^{-1} \cdot B^{-1}$, where $A$ and $B$ are invertible matrices satisfying commutative property with respect to multiplication.