Chapter 2 Matrices Ex 2.2 Solutions

From (1) and (2), we get (A + B) + C = A + (B + C).

A + B + C = 0 ∴ C = -A – B

3A – 4B + 5X = C ∴ 5X = C – 3A + 4B

A = ∴ AT = ∴ (AT)T = = A

A = ∴ AT = ∴ (AT)T = = A

Let A = Since, A is a symmetric matrix, aij = aji for all i and j

Let A = Since, A is skew-symmetric matrix,

Let A = Then AT = Since, A = AT, A is a symmetric matrix.

By equality of matrices, we get 2x + y – 1 = 3 ……..(1) and 4y = 18 ……….(2) From (2), y = Substituting y = in (1), we get 2x + – 1 = 3 ∴ 2x = 3 – = ∴ x = Hence, x = and y = .

By equality of matrices, 2a + b = 2 ….. (1) 3a – b = 3 …… (2) c + 2d = 4 …… (3) 2c – d = -1 …… (4) Adding (1) and (2), we get 5a = 5 ∴ a = 1 Substituting a = 1 in (1), we get 2(1) + b = 2 ∴ b = 0 Multiplying equation (4) by 2, we get 4c – 2d = -2 …… (5) Adding (3) and (5), we get 5c = 2 ∴ c = Substituting c = in (4), we get 2() – d = -1 ∴ d = + 1 = Hence, a = 1, b = 0, c = and d = .

The sales for July and August 2017 for Suresh and Ganesh are given by the matrices A and B as: (i) The increase in sales (in ₹) from July to August 2017 is obtained by subtracting the matrix A from B. Hence, the increase in sales (in ₹) from July to August 2017 for: Suresh book shop: ₹ 1050 in Physics, ₹ 305 in Chemistry, and ₹ 405 in Mathematics. Ganesh book shop: ₹ 350 in Physics, ₹ 445 in Chemistry, and ₹ 1295 in Mathematics. (ii) Both the book shops get 10% profit in August 2017, the profit for each bookseller in each subject in August 2017 is obtained by the scalar multiplication of matrix B by 10%, i.e. Hence, the profit for Suresh book shop are ₹ 665 in Physics, ₹ 705.50 in Chemistry and ₹ 890.50 in Mathematics and for Ganesh book shop are ₹ 700 in Physics, ₹ 750 in Chemistry and ₹ 1020 in Mathematics.