Given [(4,2)(-1,1)]M = 6I, where M is a matrix and I is the unit matrix of order 2 × 2 (i) State the order of matrix M. (ii) Find the matrix M.

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(i) Let N = Given, NM = 6I NM = Order of N = 2 × 2 Order of NM = 2 × 2 Since NM exists, we have: Number of rows of M = Number of columns in N = 2 Number of columns of M = Number of columns in NM = 2 Order of M is 2 × 2. Hence, order of M is 2 × 2. (ii) Let M = . Then, Solving for a and c : ∴ -a + c = 0 ⇒ c = a....(1) ∴ 4a + 2c = 6 ⇒ 2(2a + c = 3) ⇒ 2a + c = 3...(2) Substituting value of c from equation (1) in 2a + c = 3, we get: ⇒ 2a + a = 3 ⇒ 3a = 3 ⇒ a = ⇒ a = 1 Hence, a = c = 1 Solving for b and d: ∴ -b + d = 6 ⇒ d = b + 6...(3) ∴ 4b + 2d = 0 ⇒ 2(2b + d) = 0 ⇒ 2b + d = 0...(4) Substituting value of d from equation (3) in 2b + d = 0, we get: ⇒ 2b + b + 6 = 0 ⇒ 3b + 6 = 0 ⇒ 3b = -6 ⇒ b = ⇒ b = -2 Substituting value of b in equation(3), we get: ⇒ d = -2 + 6 ⇒ d = 4 Hence, M =

Mathematics

Given [(4,2)(-1,1)]M = 6I, where M is a matrix and I is the unit matrix of order 2 × 2 (i) State the order of matrix M. (ii) Find the matrix M.

Matrices

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