Question

If `A = [(-1, 3),(2, 0)], B = [(1, -2),(0, 3)], C = [1 - 4]` and `D = [(4), (1)]`.

1. Is the product AC possible? Justify your answer. 
2. Find the matrix X, such that X = AB + B² – DC.

Answer 1

Given: `A = [(-1, 3),(2, 0)], B = [(1, -2),(0, 3)], C = [1 - 4]` (1 × 2 row vector), `D = [(4), (1)]` (2 × 1 column vector)

Step-wise calculation: 

a. Is AC possible?

Order (A) = 2 × 2

Order (C) = 1 × 2

A × C is defined only if number of columns of A (2) = Number of rows of C (1). 

Here, 2 ≠ 1, so AC is not possible.

b. Find X = AB + B² – DC.

1. Compute AB = A × B:

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

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

= `[(-1, 11), (2, -4)]`

2. Compute B² = B × B:

B² = `[(1, -2), (0, 3)] xx [(1, -2), (0, 3)]`

= `[(11 + (-2)0, 1(-2) + (-2)3),(01 + 30, 0 xx (-2) + 3 xx 3)]`

= `[(1, -8), (0, 9)]`

3. Compute DC = D × C (2 × 1 times 1 × 2 → 2 × 2): 

DC = `[(4), (1)] xx [1, -4]`

= `[(41, 4(-4)), (11, 1(-4))]` 

= `[(4, -16), (1, -4)]`

4. Combine: AB + B² 

= `[(-1 + 1, 11 + (-8)), (2 + 0, -4 + 9)]` 

= `[(0, 3), (2, 5)]`

X = (AB + B²) – DC

= `[(0 - 4, 3 - (-16)), (2 - 1, 5 - (-4))]` 

= `[(-4, 19), (1, 9)]`