Mathematics Assignment Class 12

Chapter 3 | Matrices

Mathematics assignment for class XII chapter 3, Extra questions with answer key on chapter 3 matrices strictly according to the CBSE syllabus and very useful for examinations.

Mathematics Assignment On Matrices
Class XII Chapter 3

Strictly according to the CBSE Syllabus

Question 1

For the matrix A = $\mathbf{\left [ \begin{matrix}-3 & 6 & 0 \\4 & -5 & 8 \\0 & -7 & -2 \\\end{matrix} \right ]}$ , find $\frac{1}{2}$ (A - A'), where A' is the transpose of matrix A.

Answer:

$\mathbf{\frac{1}{2}\left [ \begin{matrix} 0& 2 & 0 \\-2 & 0 & 15 \\0 & -15 & 0 \\\end{matrix} \right ]}$

Question 2

Construct a 2 x 3 matrix whose elements in the ith row and the jth column are given by $\mathbf{a_{ij}=\frac{3i-j}{2}}$ .

Answer:

$\mathbf{\left [ \begin{matrix}1 & 1/2 & 0 \\\\5/2 & 2 & 3/2 \\\end{matrix} \right ]}$

Question 3

Find a matrix X such that 2A + B + X = 0, where

A = $\left [ \begin{matrix}-1 & 2 \\3 & 4 \\\end{matrix} \right ]$ B = $\left [ \begin{matrix}3 & -2 \\1 & 5 \\\end{matrix} \right ]$

Ans: $\left [ \begin{matrix}-1 & -2 \\-7 & -13 \\\end{matrix} \right ]$

Question 4

If A = $\left [ \begin{matrix}3 & -2 \\4 & -2 \\\end{matrix} \right ]$ , find k such that A² = KA – 2I

Ans: K = 1

Question 5

Solve for x and y ; $\left [ \begin{matrix}3 & -4 \\1 & 2 \\\end{matrix} \right ]$ $\left [ \begin{matrix}x \\y\end{matrix} \right ]$ = $\left [ \begin{matrix}3 \\11\end{matrix} \right ]$

Ans: x = 5, y = 3

Question 6

If A = $\left [ \begin{matrix}3 & -4 \\-1 & 2 \\\end{matrix} \right ]$ , find a matrix B such that AB = I

Ans: $\left [ \begin{matrix}1 & 2 \\1/2 & 3/2 \\\end{matrix} \right ]$

Question 7

If B is a skew symmetric matrix, write whether the matrix (ABA') is symmetric or skew symmetric matrix

Ans: Skew symmetric matrix

Question 8

Show that matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric matrix.

Question 9

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

Ans: $\left [ \begin{matrix}1 & 0 \\0 & 1 \\\end{matrix} \right ]$

Question 10

If A = $\left [ \begin{matrix}cosx & sinx \\-sinx & cosx \\\end{matrix} \right ]$ , find x, 0 < x < π/2 , when A + A' = I

Ans: x = π / 3

Question 11

Let A = $\left [ \begin{matrix}3 & 2 & 5 \\4 & 1 & 3 \\0 & 6 & 7 \\\end{matrix} \right ]$ , express A as sum of two matrices such as one is symmetric and other is skew symmetric.

Ansswer

$\left [ \begin{matrix}3 & 3 & 5/2 \\3 & 1 & 9/2 \\5/2 & 9/2 & 7 \\\end{matrix} \right ]+\left [ \begin{matrix}0 & -1 & 5/2 \\1 & 0 & -3/2 \\-5/2 & 3/2 & 0 \\\end{matrix} \right ]$

Question 12

If A = $\left [ \begin{matrix}2 & 0 & 1 \\2 & 1 & 3 \\1 & -1 & 0 \\\end{matrix} \right ]$ , find A² – 5A +4I

and hence find a matrix X such that A² – 5A + 4I + X = 0

Ans: $\left [ \begin{matrix}1 & 1 & 3 \\1 & 3 & 10 \\5 & -4 & -2 \\\end{matrix} \right ]$

Question 13

If A = $\left [ \begin{matrix}1 & 2 & 2 \\2 & 1 & 2 \\2 & 2 & 1 \\\end{matrix} \right ]$ , then show that A² – 4A - 5I = O and hence find A^-1

Ans: A^-1 = $\frac{1}{5}\left [ \begin{matrix}-3 & 2 & 2 \\2 & -3 & 2 \\2 & 2 & -3 \\\end{matrix} \right ]$

Question 14

Find the value of x - y ; if 2 $\left [ \begin{matrix}1 & 3 \\0 & x \\\end{matrix} \right ]$ + $\left [ \begin{matrix}y & 0 \\1 & 2 \\\end{matrix} \right ]$ = $\left [ \begin{matrix}5 & 6 \\1 & 8 \\\end{matrix} \right ]$

Answer : 0 or zero

Question 15:

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

Answer X =

$\left [ \begin{matrix}-6 & -10 \\12 & 14 \\-31 & -7 \\\end{matrix} \right ]$

Question 16

If $\left [ \begin{matrix}2 & 3 \\5 & 7 \\\end{matrix} \right ]$ $\left [ \begin{matrix}1 & -3 \\-2 & 4 \\\end{matrix} \right ]$ = $\left [ \begin{matrix}-4 & 6 \\-9 & x \\\end{matrix} \right ]$ , Find x

Ans: x = 13

Question 17

If $\left [ \begin{matrix}x & 4 & 1 \\\end{matrix} \right ]$ $\left [ \begin{matrix}2 & 1 & 2 \\1 & 0 & 2 \\0 & 2 & -4 \\\end{matrix} \right ]$ $\left [ \begin{matrix}x \\4 \\-1\end{matrix} \right ]$ = 0 , find x

Answer : x = - 2, x = - 1

Question 18

If $\left [ \begin{matrix}x & x-y \\2x+y & 7 \\\end{matrix} \right ]$ = $\left [ \begin{matrix}3 & 1 \\8 & 7 \\\end{matrix} \right ]$ , then find the value of y

Answer: y = 2

Question 19

If A = $\left [ \begin{matrix}1 & 2 & 0 \\3 & -4 & 5 \\0 & -1 & 3 \\\end{matrix} \right ]$ , then Evaluate : A²- 4A + 3I

Answer : $\left [ \begin{matrix}6 & -14 & 10 \\-21 & 36 & -25 \\-3 & 5 & -5 \\\end{matrix} \right ]$

Question 20

If A = [a_ij] is a square matrix of order 2 such that

$a_{ij}=\left\{\begin{matrix}1,\; when\; i\neq j \\0,\; when\; i=j,\end{matrix}\right.$

then find the value of A²

Answer:

$A=\left [ \begin{matrix}0 & 1 \\1 & 0 \\\end{matrix} \right ],\; A^{2}=\left [ \begin{matrix}1 &0 \\0 & 1 \\\end{matrix} \right ]$

Question: 21

$If\: A=\begin{bmatrix} 1& 3 & 0 \\-1 & 2 & 1 \\0 & 0 & 2 \\\end{bmatrix}and\: B=\begin{bmatrix}2 & 3 & 4 \\1 & 2 & 3 \\-1 & 1 & 2 \\\end{bmatrix} then\; find\; AB$

Answer :

$\begin{bmatrix}5 & 9 & 13 \\-1 & 2 & 4 \\-2 & 2 & 4 \\\end{bmatrix}$

Question 22

$If\; A=\begin{bmatrix}1 & -2 & 1 \\2 & 1 & 3 \\\end{bmatrix}and\; B=\begin{bmatrix}2 & 1 \\3 & 2 \\1 & 1 \\\end{bmatrix}$ , then find (AB)^T

Answer:

$\begin{bmatrix}-3 & 10 \\-2 & 7 \\\end{bmatrix}$

Question 23

If $3\begin{bmatrix}x & y \\z & w \\\end{bmatrix}=\begin{bmatrix}x & 6 \\-1 & 2w \\\end{bmatrix}+\begin{bmatrix} 4& x+y \\z+w & 3 \\\end{bmatrix}$ then find the value of Z

Answer: 1

Question 24

Check whether the following matrix is an orthogonal or not

$A=\begin{bmatrix}cosx & sinx \\-sinx & cosx \\\end{bmatrix}$

Answer: Yes it is an orthogonal matrix
Solution Hint
A matrix is said to be orthogonal if AA' = I

Question 25

What must be the matrix X, if $\begin{bmatrix}2 & -5 \\4 & 6 \\\end{bmatrix}+3X=\begin{bmatrix}8 & 4 \\1 & 0 \\\end{bmatrix}$

Answer: $\begin{bmatrix}2 & 3 \\-1 & -2 \\\end{bmatrix}$

Question 26

Find x and y , if A = $\left [ \begin{matrix}1 & -1 \\2 & -1 \\\end{matrix} \right ]$ , B = $\left [ \begin{matrix}x & 1 \\y & -1 \\\end{matrix} \right ]$ and (A + B)² = A² + B²

Answer : x = 1, y = 4

Solution Hint

Solve the equation (A +B)² = A² + B² we get

AB + BA = 0

Now in this equation putting the values of matrices A and B and comparing the corresponding terms we get the values of x and y.

Question 27

Verify that the matrix A = $\left [ \begin{matrix}1 & 3 & 3 \\1 & 4 & 3 \\1 & 3 & 4 \\\end{matrix} \right ]$ is the inverse of B = $\left [ \begin{matrix}7 & -3 & -3 \\-1 & 1 & 0 \\-1 & 0 & 1 \\\end{matrix} \right ]$

Solution Hint

Find the product AB, If AB = I then A is the inverse of B or B is the inverse of A

Question 28

$\mathbf{If\;\;A=\left[\begin{matrix}-1&a&2\\1&2&x\\3&1&1\\\end{matrix}\right]}$ , $\mathbf{A^{-1}=\left[\begin{matrix}1&-1&1\\-8&7&-5\\b&y&3\\\end{matrix}\right]}$

find the value of (a + x) - (b + y)

Answer: (a + x) - (b + y) = (1 + 3) - (5 - 4) = 3

Question 29

If $\mathbf{\left[\begin{matrix}a&c&0\\b&d&0\\0&0&5\\\end{matrix}\right]}$ is a scalar matrix then find the value of a + 2b + 3c + 4d

Answer: 25

Solution Hint: Scalar matrix is a matrix whose diagonal elements are equal and non diagonal elements are zero.

Question 30

If $\mathbf{\left[\begin{matrix}a&c&-1\\b&0&5\\1&-5&0\\\end{matrix}\right]}$ is a skew symmetric matrix then find the value of 2a - (b + c)

Answer: 0

Solution Hint:

For skew symmetric matrices

All diagonal elements are zero.

Corresponding Elements on the either side of the diagonal are equal but with opposite sign.

Question 31

Three schools X, Y, Z organized a fate (mela) for collecting funds for good victims in which they sold hand-held fans, mats and toys made from recycled material, the sale price of each being Rs 25, Rs 100 and Rs 50 respectively. The following table shows the number of articles of each type sold.

⟶ School

Article ↓

Hand-Held Fans

Mats

Toys

Using matrices, find the funds collected by each school by selling the above articles and the total funds collected.

Ans: Funds collected by school X = ₹ 5450

Funds collected by school Y = ₹ 5250

Funds collected by school Z = ₹ 6625

Total fund collected = ₹ 17325

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Chapter 3 | Matrices

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Mathematics Assignment On Matrices
Class XII Chapter 3