Question

If A = `[(2, 3),(0, -1)], B = [(-8),(8)]`, find a matrix M such that 2AM = B.

Answer 1

Given

A = `[(2, 3),(0, -1)]`

B = `[(-8),(8)]`

We are to find a column matrix M = `[(x),(y)]` such that,

2AM = B

Step 1: Divide both sides by 2

AM = `1/2B`

= `1/2 [(-8),(8)]`

= `[(-4),(4)]`

Step 2: Compute AM

AM = `[(2, 3),(0, -1)] xx [(x),(y)]`

= `[(2x + 3y),(-y)]`

We are told

AM = `[(-4),(4)]`

Step 3: Equate components

2x + 3y = −4   ...(1)

−y = 4

y = −4   ...(2)

Substitute y = −4 into (1):

2x + 3y = −4

2x + 3(−4) = −4

2x − 12 = −4

2x = −4 + 12

2x = 8

x = `8/2`

x = 4

The matrix M = `[(4),(-4)]`