Basic Matrix Operations

Question 1

What is a scalar multiple?

Answer 1

Denoted cA, where c is a real number (called the scalar), is a matrix obtained from A where each entry is multiplied by c.

Question 2

How are two matrices conformable for addition?

Answer 2

If, for two matrices A and B, they are both m x n matrices, then they are conformable for addition.

Question 3

How do you perform matrix addition?

Answer 3

(A + B)[i, j] = A[i, j] + B[i, j]
Pretty simple.

Question 4

List all of the possible operations on A, B, C (all with order m x n) by two scalars, α and β.
This will include both addition and scalar multiplication

Answer 4

• A = -1 * A
  A - A = O
  A + O = A
  A + B = B + A
  A + (B + C) = (A + B) + C
  α(A + B) = αA + αB
  (α + β)A = αA + βA

Question 5

Is AB = BA always true?

Answer 5

AB = BA if, and only if, A and B are both null matrices.

Question 6

How do you perform matrix multiplication?

Answer 6

For an m x n matrix A and an n x p matrix B, the matrix product C = A * B (C=AB) is defined to be an m x p matrix.

For every i in [1, …, m] and every j in [1, …, p]:
c[i ,j] = a[i, 1]b[1, j] + a[i, 2]b[2,j] + … + a[i, n]*b[n,j]

Question 7

What mathematical equation can you represent with a matrix format?

Answer 7

Linear equations!

Question 8

List all of the possible operations on A, B, C (all in correct order for which addition and multiplication is defined)

Answer 8

A(B+C) = AB + AC
(A+B)C = AC + BC
(AB)C = A(BC)

Question 9

What is (AB)^T equal to?

Answer 9

B^T * A^T