Matrix Algebra Review

Question 1

What is the commutative property of multiplication?

Does this also apply to matrix multiplication?

Answer 1

5 x 3 = 3 x 5, this is the commutative prop of multiplication,

This does not apply to matrix multiplication.
If A & B are matrices, A x B does not equal B x A

Question 2

What is an identity matrix?

Answer 2

A matrix with 1’s along the diagonal and 0’s everywhere else

Question 3

What is the associative property of multiplication?

Does this hold for matrix multiplication?

Answer 3

(3 x 5) x 2 = 3 x (5 x 2)

Yes, this holds for matrix multiplication. Matrices are not commutative but they are associative.

Question 4

What is a matrix inverse?

Do all matrices have an inverse?

Answer 4

The inverse of a matrix is a matrix that when multiplied by the original matrix results in the identity matrix

No, only square matrices (m x m dimensions) have inverses. Only those with same #rows as #columns

Question 5

What is the transpose of a matrix?

Answer 5

Let A be an m x n matrix, and let B = the transpose of A’ (transpose notation)
Then B is an n x m matrix, and Bij = Aji

Essentially, the first column in matrix A becomes the first row in A transpose