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

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

(ii) $$\left[\begin{array}{cc}{\mathbf{a}^{2}+\mathbf{b}^{2}} &{\mathbf{b}^{2}+\mathbf{c}^{2}} \\{\mathbf{a}^{2}+\mathbf{c}^{2}} & {\mathbf{a}^{2}+\mathbf{b}^{2}}\end{array}\right]+\left[\begin{array}{cc}{\mathbf{2} \mathbf{a} \mathbf{b}} & {\mathbf{2} \mathbf{b} \mathbf{c}} \\{-\mathbf{2} \mathbf{a} \mathbf{c}} & {-\mathbf{2} \mathbf{a} \mathbf{b}}\end{array}\right]$$

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

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

Question 3.
Compute the indicated products
(i) $$\left[\begin{array}{cc}{\mathbf{a}} & {\mathbf{b}} \\{-\mathbf{b}} & {\mathbf{a}}\end{array}\right]\left[\begin{array}{rr}{\mathbf{a}} & {-\mathbf{b}} \\{\mathbf{b}} & {\mathbf{a}}\end{array}\right]$$

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

(iii) $$\left[\begin{array}{cc}{1} & {-2} \\{2} & {3}\end{array}\right]\left[\begin{array}{lll}{1} & {2} & {3} \\{2} & {3} & {1}\end{array}\right]$$

(iv) $$\left[\begin{array}{lll}{2} & {3} & {4} \\{3} & {4} & {5} \\{4} & {5} & {6}\end{array}\right]\left[\begin{array}{ccc}{1} & {-3} & {5} \\{0} & {2} & {4} \\{3} & {0} & {5}\end{array}\right]$$

(v)$$\left[\begin{array}{cc}{2} & {1} \\{3} & {2} \\{-1} &{1}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {1} \\{-1} & {2} &{1}\end{array}\right]$$

(vi) $$\left[\begin{array}{ccc}{3} & {-1} & {3} \\{-1} & {0} &{2}\end{array}\right]\left[\begin{array}{cc}{2} & {-3} \\{1} & {0} \\{3} & {1}\end{array}\right]$$

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

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

Question 6.
Simplify
$$\cos \theta\left[\begin{array}{cc}{\cos \theta} & {\sin \theta} \\{-\sin \theta} & {\cos \theta}\end{array}\right]+\sin \theta\left[\begin{array}{cc}{\sin \theta} & {-\cos \theta} \\{\cos \theta} & {\sin \theta}\end{array}\right]$$

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

(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}

Question 8.
Find X,if $$\mathbf{Y}=\left[\begin{array}{ll}{3} & {2} \\{1} & {4}\end{array}\right] \text { and } 2 \mathbf{X}+\mathbf{Y}=\left[\begin{array}{rr}{\mathbf{1}} & {\mathbf{0}} \\{-3} & {\mathbf{2}}\end{array}\right]$$

Question 9.
Find X and Y,if $$2\left[\begin{array}{ll}{1} & {3} \\{0} &{x}\end{array}\right]+\left[\begin{array}{ll}{y} & {0} \\{1} & {2}\end{array}\right]=\left[\begin{array}{ll}{5} & {6} \\{1} & {8}\end{array}\right]$$

Question 10.
Solve the equation for x, y, z and t, if
$$2\left[\begin{array}{ll}{x} & {z} \\{y} & {t}\end{array}\right]+3\left[\begin{array}{cc}{1} & {-1} \\{0} & {2}\end{array}\right]=3\left[\begin{array}{cc}{3} & {5} \\{4} & {6}\end{array}\right]$$

Question 11.
$$\text { If } x\left[\begin{array}{l}{2} \\{3}\end{array}\right]+y\left[\begin{array}{c}{-1} \\{1}\end{array}\right]=\left[\begin{array}{c}{10} \\{5}\end{array}\right]$$

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.

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)

Question 14.
Show that
(i) $$\left[\begin{array}{cc}{5} & {-1} \\{6} & {7}\end{array}\right]\left[\begin{array}{cc}{2} & {1} \\{3} & {4}\end{array}\right] \neq\left[\begin{array}{cc}{2} & {1} \\{3} & {4}\end{array}\right]\left[\begin{array}{cc}{5} & {-1} \\{6} & {7}\end{array}\right]$$

(ii) $$\left[\begin{array}{ccc}{1} & {2} & {3} \\{0} & {1} & {0} \\{1} & {1} & {0}\end{array}\right]\left[\begin{array}{ccc}{-1} & {1} & {0} \\{0} & {-1} & {1} \\{2} & {3} & {4}\end{array}\right] \neq$$
$$\left[\begin{array}{ccc}{-1} & {1} & {0} \\{0} & {-1} & {1} \\{2} & {3} & {4}\end{array}\right]\left[\begin{array}{lll}{1} & {2} & {3} \\{0} & {1} & {0} \\{1} & {1} & {0}\end{array}\right]$$

Question 15.
Find
$$A^{2}-5 A+6 I, \text { If } A=\left[\begin{array}{ccc}{2} & {0} & {1} \\{2} & {1} & {3} \\{1} & {-1} & {0}\end{array}\right]$$

Question 16.
$$A^{2}-5 A+6 I, \text { If } A=\left[\begin{array}{ccc}{2} & {0} & {1} \\{2} & {1} & {3} \\{1} & {-1} & {0}\end{array}\right]$$

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

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 = $$(\mathbf{I}-\mathbf{A})\left[\begin{array}{cc}{\cos \alpha} & {-\sin \alpha} \\{\sin \alpha} & {\cos \alpha}\end{array}\right]$$

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
Let the amount invested in 1st type of bond be x then in 2nd 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.

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

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