# 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.

**M _{m x n} x M_{n x p } = M_{m x p}**

The element, **c _{ij}**, of the product matrix is obtained by multiplying every element in row

**i**of matrix

*A*by each element of column

*of matrix*

**j***B*and then adding them together.

## 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:

Calculate:

** A x B; B x A**

2.Given the matrices:

Determine if the following multiplications are possible:

1.**(A ^{t} · B) · C**

(A^{t}_{3 x 2 } · B_{2 x 2}) · C_{3 x 2 } = (A^{t} · B)_{3 x 2 } · C_{3 x 2}

The multiplication is not possible because the number of columns, (A^{t} · B ) does not coincide with the numbers of rows of C.

2.**(B · C ^{t }) · A^{t }**

(B_{2 x 2 } · C^{ t}_{2 x 3 } ) · A^{t}_{3 x 2 } = (B · C)_{2 x 3 } · A^{t}_{3 x 2 } =

=(B · C^{ t} · A^{ t} ) _{2 x 2}

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

A_{3 x 2 } · M_{m x n } · C_{3 x 2} m = 2

4Determine the dimension of **M** if **C ^{ t} · M** is a square matrix.

C^{ t}_{2 x 3} · M_{m x n} m = 3 n = 3