Question

If A = [12-23], B=[2123] and C=[-3120] verify that A(B+c) = AB+AC

Answer 1

given the context, I suspect that A,B,C are all 2x2 matrices.
Try that on for size.

Answer 2

To solve this equation, we need to perform the operations on both sides and see if they are equal:

A = [12-23]
B = [2 1 2 3]
C = [-3 1 2 0]

A(B + C) = [12 - 23]([2 1 2 3] + [-3 1 2 0])

First, we need to perform the operation inside the parentheses:

[2 1 2 3] + [-3 1 2 0] = [-1 2 4 3]

Now, we can multiply A by the sum:

[12 - 23]([-1 2 4 3]) = [-12 24 48 36 -23 -46 -92 -69]

Next, we need to calculate AB and AC:

AB = [12-23][2 1 2 3] = [24 12 24 36 -46 -23 -46 -69]

AC = [12-23][-3 1 2 0] = [-36 12 24 0 69 -23 -46 0]

Finally, we add AB and AC:

[24 12 24 36 -46 -23 -46 -69] + [-36 12 24 0 69 -23 -46 0] = [-12 24 48 36 -23 -46 -92 -69]

Therefore, A(B + C) = AB + AC, and the equation is verified.

Answer 3

Apologies for the confusion. Let's assume A, B, and C are all 2x2 matrices and verify the equation.

A = [12 -23]
B = [2 1]
[2 3]

C = [-3 1]
[2 0]

Now, let's calculate A(B + C):

B + C = [2 1] + [-3 1] = [-1 2]
[4 3]

Multiplying A by (B + C):

A(B + C) = [12 -23]([-1 2]) = [24 - 49]
[-24 -21]

Next, let's calculate AB and AC:

AB = [12 -23][2 1] = [24 - 49]
[20 - 43]

AC = [12 -23][-3 1] = [9 - 35]
[-6 2]

Finally, let's add AB and AC:

AB + AC = [24 - 49] + [9 - 35] = [33 - 84]
[14 - 41]

Hence, A(B + C) is not equal to AB + AC in this case.