2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.2

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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:

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:

(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:

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:

(ii) \(\left[\begin{array}{l}{1} \\{2} \\{3}\end{array}\right]\left[\begin{array}{lll}{2} & {3} & {4}\end{array}\right]\)
Answer:

(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:

(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:

(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:

(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:

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:

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:

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:

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:

(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:

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:

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:

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:

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:

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:

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:

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:

(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:

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:

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:

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:

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:

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 1^st type of bond be x then in 2^nd type is 30,000 – x.

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:

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:

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 = 2_x p
given n = p
∴ (B) 2 x  n is correct.