Matrix Multiplication

Question 1

What is the compatibility rule?

Answer 1

The matrix multiplication (product) AB of two matrices A and B is only defined if the number of columns of A equals the number of rows of B.

Question 2

Define matrix multiplication.

Answer 2

If A is a m×n matrix and B is a n×p matrix, then their product AB is the m×p matrix whose (i, j)-th entry is the dot product of
the i-th row of A with j-th column of B.

Question 3

What are the 4 properties of matrix multiplication?

Answer 3

1. Sometimes we can find AB, but BA is not defined (not compatible).
2. In general AB≠BA (the product is not commutative).
3. It can happen that AB=0 (zero matrix), but neither A nor B is zero
4. It can happen that AC=BC, but C≠0 (zero matrix) and A≠B (we can not cancel out C, even if C≠0).

Question 4

What are 5 other rules for Matrix multiplication?

Answer 4

Suppose A, B, C are matrices and k ∈ R. Then,
1. (AB)C = A(BC),
2. A(B + C) = AB + AC,
3. (B + C)A = BA + CA,
4. k(AB) = (kA)B = A(kB),
5. If A is a m × n matrix, then A 0n×p = 0m×p and
0q×mA = 0q×n, where 0n×p is the zero matrix of size n × p

Question 5

Define a matrix’s transpose.

Answer 5

The transpose of a m×n matrix A is the n×m matrix A^T whose rows are columns of A

Question 6

What are the 4 properties of Transpose operations?

Answer 6

1. (A+B)^T = A^T+B^T
2. (kA)^T = kA^T , k ∈ R,
3. (A^T)^T = A,
4. (AB)^T = B^T A^T (Note the order).

Question 7

What are identity matrices?.

Answer 7

Square matrices whose diagonal entries are all 1.

Question 8

What are two facts about identity matrices?

Answer 8

1. For every n×n matrix A, we have AI_n = I_nA = A.

2. For every n∈N, we have (I_n)^T = I_n

Question 9

Define block multiplication.

Answer 9

Matrices can be cut into blocks (smaller matrices) to simplify matrix multiplication.

Question 10

Describe Matrix Multiplication in Vector Format.

Answer 10

Suppose A is a m×n matrix and B a n×k matrix. We can write B in vector format as B = [b1,b2,···,bk ], where bj
is the j-th column of B and a vector in R^n. Then the product AB is the
m×k matrix given by
AB = A[b1,b2,···,bk] = [Ab1,Ab2,···,Abk].
Similarly, we can write A in vector format A=[a1,a2,···,am]^T ,
where ai
is the i-th row of A and a vector in R^n. Then, the m×k
product matrix AB is given by
AB = [a1,a2,…,am]
B = [a1B,a2B,…,amB]

Question 11

What is a fact about solving linear systems?

Answer 11

Solving [A | b] and finding x=(x1,···,xn) is the same as solving the matrix equation Ax = b. If A is the size m×n coefficient matrix, we can write A=[c1,···,cn], where ci is a vector in R^m. Then Ax=x1c1+···+xncn is linear combination of columns of A.
i.e.: Ax = b is consistent ⇔ b is a linear combination of columns of A