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Question
Determine the matrices A and B if they satisfy 2A – B + `[(6, - 6, 0),(- 4, 2, 1)]` = 0 and A – 2B = `[(3, 2, 8),(-2, 1, -7)]`
Solution
2A – B + `[(6, - 6, 0),(- 4, 2, 1)]` = 0
A – 2B = `[(3, 2, 8),(-2, 1, -7)]`
2A – B = `- [(6, - 6, 0),(- 4, 2, 1)]` ......(1)
A – 2B = `[(3, 2, 8),(-2, 1, -7)]` .......(2)
(1) ⇒ 2A – B + `[(6, - 6, 0),(- 4, 2, 1)]`
(2) × 2 2A – 4B = `[(3, 2, 8),(-2, 1, -7)]`
Substuting 0 + 3B = `[(6, - 6, 0),(- 4, 2, 1)] - 2[(3, 2, 8),(-2, 1, -7)]`
3B = `[(- 6, 6, 0),(4, - 2, -1)] - [(6, 4, 16),(-4, 2, -14)]`
= `[(-6 - 6, 6 - 4, 0 - 16),(4 + 4, - 2 - 2, - 1 + 14)]`
3B = `[(- 12, 2, - 16),(8, -4, 13)]`
B = `1/3[(- 12, 2, - 16),(8, -4, 13)]`
Substituting for B in equation (1)
(1) ⇒ `2"A" - 1/3 [(- 12, 2, - 6),(8, -4, 13)] = - [(6, 6, 0),(- 4, 2, 1)]`
2A = `1/3[(- 1, 2, 16),(8,-4, 13)] - [(6, -6, 0),(-4, 2, 1)]`
= `[((-12)/3, 2/3, (-16)/3),(8/3, (-4)/3, 13/3)] - [(6, -6, 0),(-4, 2, 1)]`
= `[(-4 - 6, 2/3 + 6, (- 16)/3 - 0),(8/3 + 4, (-4)/3 - 2, 13/3 - 1)]`
= `[(- 10, (2 + 8)/3, (-16)/3),((8 + 12)/3, (-4 - 6)/3, (13 - 3)/3)]`
2A = `[(- 10, 20/3, (-16)/3),(20/3, (- 10)/3, 10/3)]`
A = `1/2[(- 10, 0/3, (- 16)/3),(20/3, (- 10)/3, 10/3)]`
A = `[(-5, 10/3, (-8)/3),(10/3, (-5)/3, 5/3)]`
Reqired values are
A = `[(-5, 10/3, (-8)/3),(10/3, (-5)/3, 5/3)]`
B = `1/3[(- 12, 2, - 16),(8, -4, 13)]`
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