Question

recall the properties of addition you learned on page 73. Does the associative property apply when adding matrices? Give an example to support your answer.

Short answer

Yes, the associative property applies to matrix addition. For example, if A=[1, 2; 3, 4], B=[5, 6; 7, 8], and C=[9, 10; 1, 3], then (A + B) + C = A + (B + C) = [15, 18; 11, 15].

Step-by-step solution

Understanding the associative property

The associative property is an algebraic property that states for all numbers a, b, and c: (a + b) + c = a + (b + c). This property extends to the addition of matrices too.

Define Matrices A, B, and C

We'll represent matrices A, B, and C as: \n A = [1, 2; 3, 4], B=[5, 6; 7, 8] and C = [9, 10; 1, 3].

Apply the associative property

First, calculate (A + B) + C, which equals [ (1+5) + 9, (2+6) + 10; (3+7) + 1, (4+8) + 3] = [15, 18; 11, 15]. Now, calculate A + (B + C), which equals [ 1 + (5+9), 2 + (6+10); 3 + (7+1), 4 + (8+3)] = [15, 18; 11, 15]. As the results are the same, the associative property holds true for matrix addition.