1. Multiplication of Matrices
Section 2.5
Multiplication of Matrices

2. Matrix Multiplication
Size of the product: If A is a matrix of size m x n and B is a matrix of size n x p (note: the column size of A must equal the row size of B) then the product AB will be a matrix of size m x p.
Ex. Matrix A size 3 x 2 and matrix B size 2 x 5
The product AB will be a matrix of size 3 x 5
Ex. Matrix A size 3 x 4 and matrix B size 3 x 4
The product AB can’t be computed

3. For matrices A, B, and C, let AB = C.
Then
And so on…

4. Ex. Given matrices B and C find BC and CB.
Note: in general, for any two square matrices B and C

5. Laws for Matrix Multiplication
If the products and sums are defined for matrices A, B, and C we have the Associative law (AB)C = A(BC) and the Distributive law: A(B + C) = AB + AC.
Identity Matrix
The identity matrix of size n is given by
Diagonal of 1’s
In A = A and BIn = B where defined.

6. Matrix Equation Representation of a System of Linear Equations
AX = B