Multiplying Matrices

Two matrices A and B can be mulitplied together if the number of columns of A is equal to the number of rows of B.

Mm x n x Mn x p = Mm x p

The element, cij, of the product matrix is obtained by multiplying every element in row i of matrix A by each element of column j of matrix B and then adding them together.


Multiplying Matrices

Properties of Matrix Multiplication

Associative:

A · (B · C) = (A · B) · C

Multiplicative Identity

A · I = A

Where I is the identity matrix with the same order as matrix A.

Not Commutative:

A · B ≠ B · A

Distributive:

A · (B + C) = A · B + A · C

Examples

1. Given the matrices:

Matrices

Calculate:

  A x B;     B x A


Multiplying Matrices

 

Multiplying Matrices


2.Given the matrices:

Matrices

Matrices

Determine if the following multiplications are possible:

1.(At · B) · C

(At3 x 2 · B2 x 2) · C3 x 2 = (At · B)3 x 2 · C3 x 2

  The multiplication is not possible because the number of columns, (At · B ) does not coincide with the numbers of rows of C.

2.(B · Ct ) · At

(B2 x 2 · C t2 x 3 ) · At3 x 2 = (B · C)2 x 3 · At3 x 2 =

=(B · C t · A t2 x 2

3Determine the dimension of M so that the multiplication is possible: A · M · C

A3 x 2 ·  Mm x n ·  C3 x 2            m = 2

4Determine the dimension of M if C t · M is a square matrix.

  C t2 x 3 · Mm x n                     m = 3     n = 3