2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2

Students can Download Maths Chapter 3 Matrices Ex 3.2 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2

2nd PUC Maths Matrices NCERT Text Book Questions and Answers
Ex 3.2

Question 1.
\(\text { Let } \mathbf{A}=\left[\begin{array}{ll}{2} & {4} \\{3} & {2}\end{array}\right], \mathbf{B}=\left[\begin{array}{rr}{1} & {3} \\{-2} & {5}\end{array}\right], \mathbf{C}=\left[\begin{array}{rr}{-2} & {5} \\{3} & {4}\end{array}\right]\)
Find each of the following:
(i) A + B
(ii) A – B
(iii) 3A – C
(iv) AB
(v) BA
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 1
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 2

KSEEB Solutions

Question 2.
Compute the following
(i)\(\left[\begin{array}{cc}{\mathbf{a}} & {\mathbf{b}} \\{-\mathbf{b}} & {\mathbf{a}}\end{array}\right]+\left[\begin{array}{ll}{\mathbf{a}} & {\mathbf{b}} \\{\mathbf{b}} & {\mathbf{a}}\end{array}\right]\)
Answer:
\(\left[\begin{array}{cc}{2 a} & {2 b} \\{0} & {2 a}\end{array}\right]_{2 \times 2}\)

(ii) \(\left[\begin{array}{cc}{\mathbf{a}^{2}+\mathbf{b}^{2}} &{\mathbf{b}^{2}+\mathbf{c}^{2}} \\{\mathbf{a}^{2}+\mathbf{c}^{2}} & {\mathbf{a}^{2}+\mathbf{b}^{2}}\end{array}\right]+\left[\begin{array}{cc}{\mathbf{2} \mathbf{a} \mathbf{b}} & {\mathbf{2} \mathbf{b} \mathbf{c}} \\{-\mathbf{2} \mathbf{a} \mathbf{c}} & {-\mathbf{2} \mathbf{a} \mathbf{b}}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 3

(iii) \(\left[\begin{array}{ccc}{-1} & {4} & {-6} \\{8} & {5} & {16} \\{2} & {8} & {5}\end{array}\right]+\left[\begin{array}{ccc}{12} & {7} & {6} \\{8} & {0} & {5} \\{3} & {2} & {4}\end{array}\right]\)
Answer:
\(\left[\begin{array}{ccc}{11} & {11} & {0} \\{16} & {5} & {21} \\{5} & {10} & {9} \end{array}\right]_{5 \times 3}\)

(iv) \(\left[\begin{array}{cc}{\cos ^{2} x} & {\sin ^{2} x} \\{\sin ^{2} x} & {\cos ^{2} x}\end{array}\right]+\left[\begin{array}{cc}{\sin ^{2} x} & {\cos ^{2} x} \\{\cos ^{2} x} & {\sin ^{2} x}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 4
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 5

Question 3.
Compute the indicated products
(i) \(\left[\begin{array}{cc}{\mathbf{a}} & {\mathbf{b}} \\{-\mathbf{b}} & {\mathbf{a}}\end{array}\right]\left[\begin{array}{rr}{\mathbf{a}} & {-\mathbf{b}} \\{\mathbf{b}} & {\mathbf{a}}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 6

(ii) \(\left[\begin{array}{l}{1} \\{2} \\{3}\end{array}\right]\left[\begin{array}{lll}{2} & {3} & {4}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 7

(iii) \(\left[\begin{array}{cc}{1} & {-2} \\{2} & {3}\end{array}\right]\left[\begin{array}{lll}{1} & {2} & {3} \\{2} & {3} & {1}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 8

KSEEB Solutions

(iv) \(\left[\begin{array}{lll}{2} & {3} & {4} \\{3} & {4} & {5} \\{4} & {5} & {6}\end{array}\right]\left[\begin{array}{ccc}{1} & {-3} & {5} \\{0} & {2} & {4} \\{3} & {0} & {5}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 9

(v)\(\left[\begin{array}{cc}{2} & {1} \\{3} & {2} \\{-1} &{1}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {1} \\{-1} & {2} &{1}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 10

(vi) \(\left[\begin{array}{ccc}{3} & {-1} & {3} \\{-1} & {0} &{2}\end{array}\right]\left[\begin{array}{cc}{2} & {-3} \\{1} & {0} \\{3} & {1}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 11

Question 4.
\(\text { If } A=\left[\begin{array}{ccc}{1} & {2} & {-3} \\{5} & {0} & {2} \\{1} & {-1} & {1}\end{array}\right], B=\left[\begin{array}{ccc}{3} & {-1} & {2} \\{4} & {2} & {5} \\{2} & {0} & {3}\end{array}\right]\) \(\mathbf{C}=\left[\begin{array}{ccc}{4} & {1} & {2} \\{0} & {3} & {2} \\{1} & {-2} & {3}\end{array}\right]\) then compute (A + B) and
(B – C) Also ,verify that A+(B – C) = (A + B) – C
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 12
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 13

Question 5.
\(\text { If } A=\left[\begin{array}{lll}{\frac{2}{3}} & {1} & {\frac{5}{3}} \\{\frac{1}{3}} & {\frac{2}{3}} & {\frac{4}{3}} \\{\frac{7}{3}} & {2} & {\frac{2}{3}}\end{array}\right] \text { and } B=\left[\begin{array}{ccc}{\frac{2}{5}} & {\frac{3}{5}} & {1} \\{\frac{1}{5}} & {\frac{2}{5}} & {\frac{4}{5}} \\{\frac{7}{5}} & {\frac{6}{5}} & {\frac{2}{5}}\end{array}\right], \text { then }\)
compute 3A – 5B
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 14

KSEEB Solutions

Question 6.
Simplify
\(\cos \theta\left[\begin{array}{cc}{\cos \theta} & {\sin \theta} \\{-\sin \theta} & {\cos \theta}\end{array}\right]+\sin \theta\left[\begin{array}{cc}{\sin \theta} & {-\cos \theta} \\{\cos \theta} & {\sin \theta}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 15

Question 7.
Find X and Y, if
(i)
\(\mathbf{X}+\mathbf{Y}=\left[\begin{array}{ll} {7} & {\mathbf{0}} \\{\mathbf{2}} & {\mathbf{5}}\end{array}\right] \text { and } \mathbf{X}-\mathbf{Y}=\left[\begin{array}{ll}{\mathbf{3}} & {\mathbf{0}} \\{\mathbf{0}} & {\mathbf{3}}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 16
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 17

(ii)
\(\begin{aligned}2 \mathbf{X}+3 \mathbf{Y}=&\left[\begin{array}{ll}{2} & {3} \\{4} & {0}\end{array}\right] \text { and } 3 \mathbf{X}+2 \mathbf{Y} \\&=\left[\begin{array}{cc}{2} & {-2} \\{-1} & {5}\end{array}\right]\end{aligned}\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 18

KSEEB Solutions

Question 8.
Find X,if \(\mathbf{Y}=\left[\begin{array}{ll}{3} & {2} \\{1} & {4}\end{array}\right] \text { and } 2 \mathbf{X}+\mathbf{Y}=\left[\begin{array}{rr}{\mathbf{1}} & {\mathbf{0}} \\{-3} & {\mathbf{2}}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 19

Question 9.
Find X and Y,if \(2\left[\begin{array}{ll}{1} & {3} \\{0} &{x}\end{array}\right]+\left[\begin{array}{ll}{y} & {0} \\{1} & {2}\end{array}\right]=\left[\begin{array}{ll}{5} & {6} \\{1} & {8}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 20

Question 10.
Solve the equation for x, y, z and t, if
\(2\left[\begin{array}{ll}{x} & {z} \\{y} & {t}\end{array}\right]+3\left[\begin{array}{cc}{1} & {-1} \\{0} & {2}\end{array}\right]=3\left[\begin{array}{cc}{3} & {5} \\{4} & {6}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 21

Question 11.
\(\text { If } x\left[\begin{array}{l}{2} \\{3}\end{array}\right]+y\left[\begin{array}{c}{-1} \\{1}\end{array}\right]=\left[\begin{array}{c}{10} \\{5}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 22

Question 12.
Given
\(\mathbf{3}\left[\begin{array}{cc}{\mathbf{x}} & {\mathbf{y}} \\{\mathbf{z}} & {\mathbf{w}}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{x}} & {\mathbf{6}} \\{-\mathbf{1}} & {\mathbf{2} \mathbf{w}}\end{array}\right]+\left[\begin{array}{cc}{\mathbf{4}} & {\mathbf{x}+\mathbf{y}} \\{\mathbf{z}+\mathbf{w}} &{\mathbf{3}}\end{array}\right]\).
Find the value of x,y,z and w.
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 23

Question 13.
If \(\mathbf{F}(\mathbf{x})=\left[\begin{array}{ccc}{\cos \mathbf{x}} & {-\sin \mathbf{x}} & {\mathbf{0}} \\{\sin \mathbf{x}} & {\cos \mathbf{x}} & {\mathbf{0}} \\{\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}}\end{array}\right]\) F(x) F(y) = F(x +y)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 24

Question 14.
Show that
(i) \(\left[\begin{array}{cc}{5} & {-1} \\{6} & {7}\end{array}\right]\left[\begin{array}{cc}{2} & {1} \\{3} & {4}\end{array}\right] \neq\left[\begin{array}{cc}{2} & {1} \\{3} & {4}\end{array}\right]\left[\begin{array}{cc}{5} & {-1} \\{6} & {7}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 25
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 26

KSEEB Solutions

(ii) \(\left[\begin{array}{ccc}{1} & {2} & {3} \\{0} & {1} & {0} \\{1} & {1} & {0}\end{array}\right]\left[\begin{array}{ccc}{-1} & {1} & {0} \\{0} & {-1} & {1} \\{2} & {3} & {4}\end{array}\right] \neq\)
\(\left[\begin{array}{ccc}{-1} & {1} & {0} \\{0} & {-1} & {1} \\{2} & {3} & {4}\end{array}\right]\left[\begin{array}{lll}{1} & {2} & {3} \\{0} & {1} & {0} \\{1} & {1} & {0}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 27
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 28

Question 15.
Find
\(A^{2}-5 A+6 I, \text { If } A=\left[\begin{array}{ccc}{2} & {0} & {1} \\{2} & {1} & {3} \\{1} & {-1} & {0}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 29

Question 16.
\(A^{2}-5 A+6 I, \text { If } A=\left[\begin{array}{ccc}{2} & {0} & {1} \\{2} & {1} & {3} \\{1} & {-1} & {0}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 30
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 31
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 32

KSEEB Solutions

Question 17.
If \(A=\left[\begin{array}{ll}{3} & {-2} \\{4} & {-2}\end{array}\right] \text { and } I=\left[\begin{array}{ll}{1} & {0} \\{0} & {1}\end{array}\right]\) ,find k so that
A2 = KA – 21
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 33
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 34

Question 18.
If \(\mathbf{A}=\left[\begin{array}{cc}{0} & {-\tan \frac{\alpha}{2}} \\{\tan \frac{\alpha}{2}} & {0}\end{array}\right]\) and I is the identity matrix of order 2, show that I + A = \((\mathbf{I}-\mathbf{A})\left[\begin{array}{cc}{\cos \alpha} & {-\sin \alpha} \\{\sin \alpha} & {\cos \alpha}\end{array}\right]\)
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 35
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 36

Question 19.
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 fund must obtain an annual total interest of:
(a) ₹ 1800
(b) ₹ 2000
Answer:
Let the amount invested in 1st type of bond be x then in 2nd type is 30,000 – x.
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 37
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 38

KSEEB Solutions

Question 20.
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 x n, 3 x k, 2 x p, n x 3 and p x k, respectively. Choose the correct answer in Exercises 21 and 22.
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 39
Assume X, Y, Z, W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3 and p x k respectively. Choose the correct answer in exercise 21 and 22.

Question 21.
The restriction on n, k and p so that PY + WY willbe defined are:
(A) k = 3, p = n
(B) k is arbitrary, p = 2
(C) p is arbitrary, k = 3
(D) k = 2,p = 3
Answer:
2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2 40

Question 22.
If n = p, then the order of the matrix 7X – 5Z is:
(A) p x 2
(B) 2 x n
(C) n x 3
(D) p x n
Answer:
X is of order 2 x  n
Z is of order 2 x p
2 x n = 2x p
given n = p
∴ (B) 2 x  n is correct.

 

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