Matrices Class 12

In the matrix $\mathrm{A}=\left[\begin{array}{cccc}2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17\end{array}\right]$, write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Construct a $2 \times 2$ matrix, $\mathrm{A}=\left[a_{i j}\right]$, whose elements are given by:
(i) $a_{i j}=\frac{(i+j)^2}{2}$

Construct a $2 \times 2$ matrix, $\mathrm{A}=\left[a_{i j}\right]$, whose elements are given by:
(ii) $a_{i j}=\frac{i}{j}$

Construct a $2 \times 2$ matrix, $\mathrm{A}=\left[a_{i j}\right]$, whose elements are given by:
(iii) $a_{i j}=\frac{(i+2 j)^2}{2}$

Construct a 3 × 4 matrix, whose elements are given by:
(i) $a_{i j}=\frac{1}{2}|-3 i+j|$

Construct a 3 × 4 matrix, whose elements are given by:
(ii) $a_{i j}=2 i-j$

Find the values of $x, y$ and $z$ from the following equations:
(i) $\left[\begin{array}{ll}4 & 3 \\ x & 5\end{array}\right]=\left[\begin{array}{ll}y & z \\ 1 & 5\end{array}\right]$

Find the values of $x, y$ and $z$ from the following equations:
(ii) $\left[\begin{array}{cc}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{ll}6 & 2 \\ 5 & 8\end{array}\right]$

Find the values of $x, y$ and $z$ from the following equations:
(iii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$

Find the value of $a, b, c$ and $d$ from the equation:

$$
\left[\begin{array}{cc}
a-b & 2 a+c \\
2 a-b & 3 c+d
\end{array}\right]=\left[\begin{array}{cc}
-1 & 5 \\
0 & 13
\end{array}\right]
$$

\begin{itemize}\item[8.] $\mathrm{A}=\left[a_{i j}\right]_{m \times n}$ is a square matrix, if
(A) $m<n$
(B) $m>n$
(C) $m=n$
(D) None of these

Which of the given values of $x$ and $y$ make the following pair of matrices equal $\left[\begin{array}{cc}3 x+7 & 5 \\ y+1 & 2-3 x\end{array}\right],\left[\begin{array}{cc}0 & y-2 \\ 8 & 4\end{array}\right]$
(A) $x=\frac{-1}{3}, y=7$
(B) Not possible to find
(C) $y=7, \quad x=\frac{-2}{3}$
(D) $x=\frac{-1}{3}, y=\frac{-2}{3}$

The number of all possible matrices of order $3 \times 3$ with each entry 0 or 1 is:
(A) 27
(B) 18
(C) 81
(D) 512

Let $\mathrm{A}=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], \mathrm{B}=\left[\begin{array}{rr}1 & 3 \\ -2 & 5\end{array}\right], \mathrm{C}=\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$
Find each of the following:
(i) $\mathrm{A}+\mathrm{B}$
(ii) $\mathrm{A}-\mathrm{B}$
(iii) 3A - C
(iv) AB
(v) BA

Compute the following:
(i) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]+\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$

Compute the following:
(ii) $\left[\begin{array}{ll}a^2+b^2 & b^2+c^2 \\ a^2+c^2 & a^2+b^2\end{array}\right]+\left[\begin{array}{rr}2 a b & 2 b c \\ -2 a c & -2 a b\end{array}\right]$

Compute the following:
(iii) $\left[\begin{array}{rrr}-1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5\end{array}\right]+\left[\begin{array}{ccc}12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4\end{array}\right]$

Compute the following:
(iv) $\left[\begin{array}{cc}\cos ^2 x & \sin ^2 x \\ \sin ^2 x & \cos ^2 x\end{array}\right]+\left[\begin{array}{cc}\sin ^2 x & \cos ^2 x \\ \cos ^2 x & \sin ^2 x\end{array}\right]$

Compute the indicated products.
(i) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]$

Compute the indicated products.
(ii) $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\left[\begin{array}{lll}2 & 3 & 4\end{array}\right]$

Compute the indicated products.
(iii) $\left[\begin{array}{rr}1 & -2 \\ 2 & 3\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right]$

Compute the indicated products.
(iv) $\left[\begin{array}{lll}2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6\end{array}\right]\left[\begin{array}{rrr}1 & -3 & 5 \\ 0 & 2 & 4 \\ 3 & 0 & 5\end{array}\right]$

Compute the indicated products.
(v) $\left[\begin{array}{rr}2 & 1 \\ 3 & 2 \\ -1 & 1\end{array}\right]\left[\begin{array}{rrr}1 & 0 & 1 \\ -1 & 2 & 1\end{array}\right]$

Compute the indicated products.
(vi) $\left[\begin{array}{rrr}3 & -1 & 3 \\ -1 & 0 & 2\end{array}\right]\left[\begin{array}{rr}2 & -3 \\ 1 & 0 \\ 3 & 1\end{array}\right]$

If $\mathrm{A}=\left[\begin{array}{rrr}1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1\end{array}\right], \mathrm{B}=\left[\begin{array}{rrr}3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3\end{array}\right]$ and $\mathrm{C}=\left[\begin{array}{rrr}4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3\end{array}\right]$, then compute $(\mathrm{A}+\mathrm{B})$ and $(\mathrm{B}-\mathrm{C})$. Also, verify that $\mathrm{A}+(\mathrm{B}-\mathrm{C})=(\mathrm{A}+\mathrm{B})-\mathrm{C}$.

If $\mathrm{A}=\left[\begin{array}{ccc}\frac{2}{3} & 1 & \frac{5}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{7}{3} & 2 & \frac{2}{3}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}\frac{2}{5} & \frac{3}{5} & 1 \\ \frac{1}{5} & \frac{2}{5} & \frac{4}{5} \\ \frac{7}{5} & \frac{6}{5} & \frac{2}{5}\end{array}\right]$, then compute $3 A-5 B$.

Simplify $\cos \theta\left[\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{rr}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]$

Find $X$ and $Y$, if
(i) $\mathrm{X}+\mathrm{Y}=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$ and $\mathrm{X}-\mathrm{Y}=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

Find $X$ and $Y$, if
(ii) $2 \mathrm{X}+3 \mathrm{Y}=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$ and $3 \mathrm{X}+2 \mathrm{Y}=\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right]$

Find X , if $\mathrm{Y}=\left[\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right]$ and $2 \mathrm{X}+\mathrm{Y}=\left[\begin{array}{rr}1 & 0 \\ -3 & 2\end{array}\right]$

Find $x$ and $y$, if $2\left[\begin{array}{cc}1 & 3 \\ 0 & x\end{array}\right]+\left[\begin{array}{cc}y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}5 & 6 \\ 1 & 8\end{array}\right]$

Solve the equation for $x, y, z$ and $t$, if $2\left[\begin{array}{cc}x & z \\ y & t\end{array}\right]+3\left[\begin{array}{rr}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$

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

Given $3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]$, find the values of $x, y, z$ and $w$.

If $\mathrm{F}(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$, show that $\mathrm{F}(x) \mathrm{F}(y)=\mathrm{F}(x+y)$.

Show that
(i) $\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right] \neq\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]\left[\begin{array}{rr}5 & -1 \\ 6 & 7\end{array}\right]$
(ii) $\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right] \neq\left[\begin{array}{rrr}-1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0\end{array}\right]$

Find $A^2-5 A+6 I$, if $A=\left[\begin{array}{rrr}2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right]$

If $\mathrm{A}=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]$, prove that $\mathrm{A}^3-6 \mathrm{~A}^2+7 \mathrm{~A}+2 \mathrm{I}=0$

If $\mathrm{A}=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$ and $\mathrm{I}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, find $k$ so that $\mathrm{A}^2=k \mathrm{~A}-2 \mathrm{I}$

If $\mathrm{A}=\left[\begin{array}{cc}0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0\end{array}\right]$ and I is the identity matrix of order 2 , show that
$$
I+A=(I-A)\left[\begin{array}{rr}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{array}\right]
$$

A trust fund has ₹ 30,000 that must be invested in two different types of bonds. The first bond pays $5 \%$ interest per year, and the second bond pays $7 \%$ interest per year. Using matrix multiplication, determine how to divide ₹ 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
(a) ₹1800
(b) ₹2000

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are ₹ 80 , ₹ 60 and ₹ 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Assume X, Y, Z, W and P are matrices of order $2 \times n, 3 \times k, 2 \times p, n \times 3$ and $p \times k$, respectively. Choose the correct answer.
The restriction on $n, k$ and $p$ so that $\mathrm{PY}+\mathrm{WY}$ will be defined are:
(A) $k=3, p=n$
(B) $k$ is arbitrary, $p=2$
(C) $p$ is arbitrary, $k=3$
(D) $k=2, p=3$

If $n=p$, then the order of the matrix $7 \mathrm{X}-5 \mathrm{Z}$ is:
(A) $p \times 2$
(B) $2 \times n$
(C) $n \times 3$
(D) $p \times n$

Find the transpose of each of the following matrices:
(i) $\left[\begin{array}{c}5 \\ \frac{1}{2} \\ -1\end{array}\right]$
(ii) $\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]$
(iii) $\left[\begin{array}{ccc}-1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1\end{array}\right]$

If $\mathrm{A}=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right]$, then verify that
(i) $(\mathrm{A}+\mathrm{B})^{\prime}=\mathrm{A}^{\prime}+\mathrm{B}^{\prime}$,
(ii) $(\mathrm{A}-\mathrm{B})^{\prime}=\mathrm{A}^{\prime}-\mathrm{B}^{\prime}$

If $A^{\prime}=\left[\begin{array}{rr}3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right]$, then verify that
(i) $(\mathrm{A}+\mathrm{B})^{\prime}=\mathrm{A}^{\prime}+\mathrm{B}^{\prime}$
(ii) $(\mathrm{A}-\mathrm{B})^{\prime}=\mathrm{A}^{\prime}-\mathrm{B}^{\prime}$

If $A^{\prime}=\left[\begin{array}{cc}-2 & 3 \\ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{rr}-1 & 0 \\ 1 & 2\end{array}\right]$, then find $(A+2 B)^{\prime}$

For the matrices A and B , verify that $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}$, where
(i) $\mathrm{A}=\left[\begin{array}{r}1 \\ -4 \\ 3\end{array}\right], \mathrm{B}=\left[\begin{array}{lll}-1 & 2 & 1\end{array}\right]$

For the matrices A and B , verify that $(\mathrm{AB})^{\prime}=\mathrm{B}^{\prime} \mathrm{A}^{\prime}$, where
(ii) $\mathrm{A}=\left[\begin{array}{l}0 \\ 1 \\ 2\end{array}\right], \mathrm{B}=\left[\begin{array}{lll}1 & 5 & 7\end{array}\right]$

$\mathrm{A}=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$, then verify that $\mathrm{A}^{\prime} \mathrm{A}=\mathrm{I}$

If $\mathrm{A}=\left[\begin{array}{cc}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right]$, then verify that $\mathrm{A}^{\prime} \mathrm{A}=\mathrm{I}$

Show that the matrix $\mathrm{A}=\left[\begin{array}{rrr}1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3\end{array}\right]$ is a symmetric matrix.

Show that the matrix $\mathrm{A}=\left[\begin{array}{rrr}0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0\end{array}\right]$ is a skew symmetric matrix.

For the matrix $\mathrm{A}=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right]$, verify that
(i) $\left(\mathrm{A}+\mathrm{A}^{\prime}\right)$ is a symmetric matrix
(ii) $\left(\mathrm{A}-\mathrm{A}^{\prime}\right)$ is a skew symmetric matrix

Find $\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)$ and $\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right)$, when $\mathrm{A}=\left[\begin{array}{rrr}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
$\left[\begin{array}{rr}3 & 5 \\ 1 & -1\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
$\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
$\left[\begin{array}{rrr}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
$\left[\begin{array}{rr}1 & 5 \\ -1 & 2\end{array}\right]$

Choose the correct answer.
If $\mathrm{A}, \mathrm{B}$ are symmetric matrices of same order, then $\mathrm{AB}-\mathrm{BA}$ is a
(A) Skew symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(D) Identity matrix

If $\mathrm{A}=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$, and $\mathrm{A}+\mathrm{A}^{\prime}=\mathrm{I}$, then the value of $\alpha$ is
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{3}$
(C) $\pi$
(D) $\frac{3 \pi}{2}$

Matrices A and B will be inverse of each other only if
(A) $\mathrm{AB}=\mathrm{BA}$
(B) $\mathrm{AB}=\mathrm{BA}=0$
(C) $\mathrm{AB}=0, \mathrm{BA}=\mathrm{I}$
(D) $\mathrm{AB}=\mathrm{BA}=\mathrm{I}$

Consider the following information regarding the number of men and women workers in three factories I, II and III

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

If a matrix has 8 elements, what are the possible orders it can have?

Construct a $3 \times 2$ matrix whose elements are given by $a_{i j}=\frac{1}{2}|i-3 j|$.

If $\left[\begin{array}{ccc}x+3 & z+4 & 2 y-7 \\ -6 & a-1 & 0 \\ b-3 & -21 & 0\end{array}\right]=\left[\begin{array}{ccc}0 & 6 & 3 y-2 \\ -6 & -3 & 2 c+2 \\ 2 b+4 & -21 & 0\end{array}\right]$
Find the values of $a, b, c, x, y$ and $z$.

Find the values of $a, b, c$, and $d$ from the following equation:

$$
\left[\begin{array}{cc}
2 a+b & a-2 b \\
5 c-d & 4 c+3 d
\end{array}\right]=\left[\begin{array}{cc}
4 & -3 \\
11 & 24
\end{array}\right]
$$

Given $\mathrm{A}=\left[\begin{array}{ccc}\sqrt{3} & 1 & -1 \\ 2 & 3 & 0\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}2 & \sqrt{5} & 1 \\ -2 & 3 & \frac{1}{2}\end{array}\right]$, find $\mathrm{A}+\mathrm{B}$

If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}3 & -1 & 3 \\ -1 & 0 & 2\end{array}\right]$, then find $2 A-B$.

If $A=\left[\begin{array}{rr}8 & 0 \\ 4 & -2 \\ 3 & 6\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -2 \\ 4 & 2 \\ -5 & 1\end{array}\right]$, then find the matrix $X$, such that $2 \mathrm{~A}+3 \mathrm{X}=5 \mathrm{~B}$.

Find X and Y , if $\mathrm{X}+\mathrm{Y}=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$ and $\mathrm{X}-\mathrm{Y}=\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$.

Find the values of $x$ and $y$ from the following equation:

$$
2\left[\begin{array}{cc}
x & 5 \\
7 & y-3
\end{array}\right]+\left[\begin{array}{cr}
3 & -4 \\
1 & 2
\end{array}\right]=\left[\begin{array}{cc}
7 & 6 \\
15 & 14
\end{array}\right]
$$

Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.

September Sales (in Rupees)
$$
\mathrm{A}=\left[\begin{array}{ccc}
\text { Basmati } & \text { Permal } & \text { Naura } \\
10,000 & 20,000 & 30,000 \\
50,000 & 30,000 & 10,000
\end{array}\right] \begin{aligned}
& \text { Ramkishan } \\
& \text { Gurcharan Singh }
\end{aligned}
$$
October Sales (in Rupees)
$$
B=\left[\begin{array}{ccc}
\text { Basmati } & \text { Permal } & \text { Naura } \\
5000 & 10,000 & 6000 \\
20,000 & 10,000 & 10,000
\end{array}\right] \begin{aligned}
& \text { Ramkishan } \\
& \text { Gurcharan Singh }
\end{aligned}
$$
[(i) Find the combined sales in September and October for each farmer in each variety.
(ii) Find the decrease in sales from September to October.
(iii) If both farmers receive 2\% profit on gross sales, compute the profit for each farmer and for each variety sold in October.

Find AB , if $\mathrm{A}=\left[\begin{array}{ll}6 & 9 \\ 2 & 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{lll}2 & 6 & 0 \\ 7 & 9 & 8\end{array}\right]$.

If $\mathrm{A}=\left[\begin{array}{rrr}1 & -2 & 3 \\ -4 & 2 & 5\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}2 & 3 \\ 4 & 5 \\ 2 & 1\end{array}\right]$, then find $\mathrm{AB}, \mathrm{BA}$. Show that $A B \neq B A$.

If $\mathrm{A}=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$, then $\mathrm{AB}=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]$.
and

$$
\mathrm{BA}=\left[\begin{array}{rr}
0 & -1 \\
1 & 0
\end{array}\right] . \text { Clearly } \mathrm{AB} \neq \mathrm{BA} .
$$

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

If $\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2\end{array}\right], \mathrm{B}=\left[\begin{array}{rr}1 & 3 \\ 0 & 2 \\ -1 & 4\end{array}\right]$ and $\mathrm{C}=\left[\begin{array}{cccc}1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1\end{array}\right]$, find $\mathrm{A}(\mathrm{BC}),(\mathrm{AB}) \mathrm{C}$ and show that $(\mathrm{AB}) \mathrm{C}=\mathrm{A}(\mathrm{BC})$.

If $\mathrm{A}=\left[\begin{array}{rrr}0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0\end{array}\right], \mathrm{B}=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right], \mathrm{C}=\left[\begin{array}{r}2 \\ -2 \\ 3\end{array}\right]$
Calculate AC, BC and $(\mathrm{A}+\mathrm{B}) \mathrm{C}$. Also, verify that $(\mathrm{A}+\mathrm{B}) \mathrm{C}=\mathrm{AC}+\mathrm{BC}$

If $A=\left[\begin{array}{rrr}1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1\end{array}\right]$, then show that $A^3-23 A-40 I=0$

In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix A as

$$
\mathrm{A}=\left[\begin{array}{c}
\text { Cost per contact } \\
40 \\
100 \\
50
\end{array}\right] \begin{aligned}
& \text { Telephone } \\
& \text { Housecall } \\
& \text { Letter }
\end{aligned}
$$


The number of contacts of each type made in two cities X and Y is given by
Telephone Housecall Letter
$B=\left[\begin{array}{ccc}1000 & 500 & 5000 \\ 3000 & 1000 & 10,000\end{array}\right] \rightarrow X$. Find the total amount spent by the group in the two cities X and Y .

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

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

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

If $\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$, then prove that $\mathrm{A}^n=\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right], n \in \mathbf{N}$.

If A and B are symmetric matrices of the same order, then show that AB is symmetric if and only if A and B commute, that is $\mathrm{AB}=\mathrm{BA}$.

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

If A and B are symmetric matrices, prove that AB - BA is a skew symmetric matrix.

Show that the matrix $\mathrm{B}^{\prime} \mathrm{AB}$ is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Find the values of $x, y, z$ if the matrix $\mathrm{A}=\left[\begin{array}{ccr}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]$ satisfy the equation
$$
\mathrm{A}^{\prime} \mathrm{A}=\mathrm{I} .
$$

For what values of $x$ : [1 2 1 1 ] $\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=\mathrm{O}$ ?

If $\mathrm{A}=\left[\begin{array}{rr}3 & 1 \\ -1 & 2\end{array}\right]$, show that $\mathrm{A}^2-5 \mathrm{~A}+7 \mathrm{I}=0$.

Find $x$, if $\left[\begin{array}{lll}x & -5 & -1\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$

A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below:

(a) If unit sale prices of $x, y$ and $z$ are ₹ 2.50, ₹ 1.50 and ₹ 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are ₹ 2.00 , ₹ 1.00 and 50 paise respectively. Find the gross profit.

Find the matrix X so that $\mathrm{X}\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right]=\left[\begin{array}{rrr}-7 & -8 & -9 \\ 2 & 4 & 6\end{array}\right]$

If $\mathrm{A}=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha\end{array}\right]$ is such that $\mathrm{A}^2=\mathrm{I}$, then
(A) $1+\alpha^2+\beta \gamma=0$
(B) $1-\alpha^2+\beta \gamma=0$
(C) $1-\alpha^2-\beta \gamma=0$
(D) $1+\alpha^2-\beta \gamma=0$

If the matrix A is both symmetric and skew symmetric, then
(A) A is a diagonal matrix
(B) A is a zero matrix
(C) A is a square matrix
(D) None of these

If A is square matrix such that $\mathrm{A}^2=\mathrm{A}$, then $(\mathrm{I}+\mathrm{A})^3-7 \mathrm{~A}$ is equal to
(A) A
(B) $\mathrm{I}-\mathrm{A}$
(C) I
(D) 3A