Matrices — Practice

Question 1

If A = [[2, 3], [1, −1]] and B = [[1, 0], [2, 1]], find AB and BA. Comment on commutativity.

Accepted Answer

AB: A(2×2) × B(2×2) → 2×2. AB = [[2×1+3×2, 2×0+3×1], [1×1+(−1)×2, 1×0+(−1)×1]] = [[2+6, 0+3], [1−2, 0−1]] = [[8, 3], [−1, −1]]. BA: B(2×2) × A(2×2) → 2×2. BA = [[1×2+0×1, 1×3+0×(−1)], [2×2+1×1, 2×3+1×(−1)]] = [[2, 3], [4+1, 6−1]] = [[2, 3], [5, 5]]. AB = [[8,3],[−1,−1]] ≠ BA = [[2,3],[5,5]]. This confirms that MATRIX MULTIPLICATION IS NOT COMMUTATIVE in general (AB ≠ BA).

Question 2

Express A = [[3, −2, 1], [4, 0, 5], [−1, 2, 3]] as the sum of a symmetric and a skew-symmetric matrix.

Accepted Answer

SYMMETRIC PART: P = ½(A + A'). A' = [[3, 4, −1], [−2, 0, 2], [1, 5, 3]]. A + A' = [[6, 2, 0], [2, 0, 7], [0, 7, 6]]. P = ½(A + A') = [[3, 1, 0], [1, 0, 7/2], [0, 7/2, 3]]. Verify: P' = P ✓ (symmetric). SKEW-SYMMETRIC PART: Q = ½(A − A'). A − A' = [[0, −6, 2], [6, 0, 3], [−2, −3, 0]]. Q = ½(A − A') = [[0, −3, 1], [3, 0, 3/2], [−1, −3/2, 0]]. Verify: Q' = −Q ✓ and diagonal elements = 0 ✓ (skew-symmetric). CHECK: P + Q = [[3,1,0],[1,0,7/2],[0,7/2,3]] + [[0,−3,1],[3,0,3/2],[−1,−3/2,0]] = [[3,−2,1],[4,0,5],[−1,2,3]] = A ✓

Question 3

Using elementary row operations, find the inverse of A = [[1, 2, 3], [2, 5, 7], [−2, −4, −5]].

Accepted Answer

Write [A | I]: [[1,2,3 | 1,0,0], [2,5,7 | 0,1,0], [−2,−4,−5 | 0,0,1]]. R₂ → R₂ − 2R₁: [[1,2,3 | 1,0,0], [0,1,1 | −2,1,0], [−2,−4,−5 | 0,0,1]]. R₃ → R₃ + 2R₁: [[1,2,3 | 1,0,0], [0,1,1 | −2,1,0], [0,0,1 | 2,0,1]]. R₂ → R₂ − R₃: [[1,2,3 | 1,0,0], [0,1,0 | −4,1,−1], [0,0,1 | 2,0,1]]. R₁ → R₁ − 3R₃: [[1,2,0 | −5,0,−3], [0,1,0 | −4,1,−1], [0,0,1 | 2,0,1]]. R₁ → R₁ − 2R₂: [[1,0,0 | 3,−2,−1], [0,1,0 | −4,1,−1], [0,0,1 | 2,0,1]]. A is now reduced to I, so: A⁻¹ = [[3,−2,−1],[−4,1,−1],[2,0,1]]. VERIFY: A × A⁻¹ should = I (student should check at least one row).

Practice — Matrices

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