Question

If A = `[(1, 4), (1, -3)]` and B = `[(1, 2),(-1, -1)]`, find:

1. (A + B)²
2. A² + B²
3. Is (A + B)² = A² + B² ?

Answer 1

i.

A + B = `[(1, 4),(1, -3)] + [(1, 2),(-1, -1)]`

= `[((1 + 1, 4 + 2)),((1 - 1, - 3 - 1))]`

= `[(2, 6),(0, -4)]`

 Now, (A + B)² = (A + B)(A + B)

= `[(2, 6),(0, -4)]  [(2, 6),(0, -4)]`

= `[((2)(2) + (6)(0), (2)(6) + (6)(-4)), ((0)(2) + (-4)(0), (0)(6) + (-4)(-4))]`

= `[(4, -12),(0, 16)]`

ii. 

A² = `[(1, 4), (1, -3)][(1, 4), (1, -3)]`

= `[(1 + 4, 4 - 12),(1 - 3, 4 + 9)]`

= `[(5, -8),(-2, 13)]`

B² = `[(1, 2),(-1, -1)][(1,2),(-1, -1)]`

`[(1 -2, 2 - 2),(-1 + 1, -2 + 1)]`

`[(-1, 0),(0, -1)]`

A² + B² = `[(5, -8),(-2, 13)] +  [(-1, 0),(0, -1)]`

A² + B² = `[(4, -8),(-2, 12)]`

iii. No, (A + B)² ≠ A² + B².