Question

If A = `[(2, 3),(0, 5)], B = [(-7, -3),(2, 4)] and C = [(-1, 0),(-4, 7)]`, verify that (A − 2B) + 3C = A – (2B − 3C).

Answer 1

Compute (A − 2B) + 3C

Find 2B

2B = `2[(-7, -3),(2, 4)]`

= `[(2(-7), 2(-3)),(2(2), 2(4))]`

= `[(-14, -6),(4, 8)]`

Find A − 2B

A − 2B = `[(2, 3),(0, 5)] - [(- 14, -6),(4, 8)]`

= `[(2 - (-14), 3 - (-6)),(0 - 4, 5 - 8)]`

= `[(2 + 14, 3 + 6),(-4, -3)]`

= `[(16, 9),(-4, -3)]`

Find 3C

3C = `3[(-1, 0),(-4, 7)]`

= `[(3(-1), 3(0)),(3(-4), 3(7))]`

= `[(-3, 0),(-12, 21)]`

Find (A − 2B) + 3C

(A − 2B) + 3C = `[(16, 9),(-4, -3)] + [(-3, 0),(-12, 21)]`

= `[(16 + (-3), 9 + 0),(-4 + (-12), -3 + 21)]`

= `[(13, 9),(-16, 18)]`

Compute A – (2B − 3C)

Find 2B − 3C

2B − 3C = `[(-14, -6),(4, 8)] - [(-3, 0),(-12, 21)]`

= `[(-14 - (-3), -6 - 0),(4 - (-12), 8 - 21)]`

= `[(-11, -6),(16, -13)]`

Find A − (2B − 3C)

A − (2B − 3C) = `[(2, 3),(0, 5)] - [(-11, -6),(16, -13)]`

= `[(2 - (-11), 3 - (-6)),(0 - 16, 5 - (-13))]`

= `[(13, 9),(-16, 18)]`

(A − 2B) + 3C = `[(13, 9),(-16, 18)]`

A − (2B − 3C) = `[(13, 9),(-16, 18)]`

Hence verified,

(A − 2B) + 3C = A − (2B − 3C)