Multiplication of matrices.

a) Multiplication by a scalar (a number).

Consider the matrix

image 51

Then;

image 52

This is the same as multiplying each entry of M by 5.ie

image 53

In general therefore, to multiply a matrix by a real number (a scalar); we multiply each element in the matrix by the number.


Example

Given that matrix

image 54
image 55

Example

Given that;

image 56

Find:

image 57

Solution

image 58

b) Multiplication of two or more matrices To multiply two or more matrices together, we multiply the first number in the row matrix by the first number in the column matrix, the second number in the row matrix by the second number in the column matrix and so on and then add the products together.

NB:
For multiplication of two matrices to be possible, it is essential that the number of columns in the first matrix should be the same as thenumber of rows in the second matrix.

Given that mattrix: A

image 59

Find;

i) AB

ii) BA

Solutions

image 60

This matrix multiplication is not compatible i.e. multiplication in this case is impossible because the number of columns in the first matrix is
not equal to the number of rows in the second matrix. Generally, for any two matrices A and B; AB ≠ BA except when one of the matrices is an identity matrix.

Example
Work out the following

image 61

solutions

image 62