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Matrix Algebra: Basic Matrix Laws

1. Given the matrices A, B and C, compute:

(a) AC + BC (It is much faster if you use the distributive law for matrices first.)
(b) 2A - 3A
(c) Perform the Boolean Product operation on the following zero-one matrices.

Please refer to the attachment for the complete question.

2. We know that matrix algebra behaves similar to (but not exactly the same as) regular algebra. The statements in parts a and b illustrate a couple of the differences between the two structures.

Let A and B be arbitrary n x n matrices whose entries are real numbers.

(a) Use basic matrix laws only to expand (A + B)^2. Explain all steps.
(b) Is (A - B)(A + B) = A^2 - B^2 ? Explain as you did in part (a).

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Solution Preview

Please refer to the attachment for detailed answers.

1 (a)
Using distributive law for matrices, we can rewrite "AC + BC" as "(A + B)C". This will reduce two matrix multiplication and one matrix addition operations to one matrix multiplication and one matrix addition operations.

(A + B)
= | 0 0 0 |
| 0 3 0 |
| 1 0 1 |

Thus,
(A + B)C
= | 0 0 |
|-3 0 |
| 3 -1 |

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