# Further Mathematics - Exam Revision ## Operations with Matrices

Authored By Carmen Popescu-Rose 0 Comment(s)

### Addition and subtraction of matrices

Requirement: Matrices must be of the same order.

How to: Corresponding elements are added (in addition) or subtracted (in subtraction).

Example

Consider the two matrices given below.  a. Calculate the matrix AB.

The two matrices are of the same order, (2 × 3),  so they can be added.

b. Calculate the matrix A − B.

The two matrices are of the same order, (2 × 3),  so they can be subtracted. ### Multiplying a matrix by a scalar

A scalar is any real number.

When multiplying a matrix by a scalar, each element of the matrix is multiplied by the scalar.

Example

Calculate the matrix 2A. ### Multiplying matrices

Requirement: Matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix.

Example

The order of matrix M is (a × b), where a and b are positive integers.

The order of matrix N is (b × c), where b and c are positive integers.

The product MN is defined because the number of columns in matrix M is equal to the number of rows in the matrix N.

(a × b)(b × c)

The order of the product matrix is (a × c).

The product MN is not always equal to the product NM.

How to: This can be easily explained using an example.

Example

Consider the two matrices given below. and The order of the matrix M is (2 × 3), where a and b are positive integers.

The order of the matrix N is (3 × 2), where b and c are positive integers.

The product MN is defined because the number of columns in matrix M is equal to the number of rows in the matrix N.

(2 × 3) (3 × 2)

The order of the product matrix is (2 × 2).

Method

Step 1. Multiply the elements in the first row of matrix M by the elements in the first column of matrix N, in pairs.  2 × 4 + 0 × (−8) + (−1) × (−3)

Step 2. Add up all the products.

= 80 + 3

= 11

Step 3. Write this answer in row 1 column 1 of a (2 × 2) matrix. Step 4. Repeat Steps 1 − 3 for the multiplication of row 1 and column 2.  2 × (−5) + 0 × 1 + (−1) × 6 = −10 + 0 + (−6)                                               = − 16

Step 5. Write this answer in row 1 column 2 in the (2 × 2) matrix. Step 6. Repeat Steps 1 − 3 for the multiplication of row 2 and column 1.  7 × 4 + (−3) × (−8) + 6 × (−3) = 28 + 24 + (−18)                                                    = 34

Step 7. Write this answer in row 2 column 1 in the (2 × 2) matrix. Step 8. Repeat Steps 1 − 3 for the multiplication of row 2 and column 2.  7 × (−5) + (−3) × 1 + 6 × 6 = (−35) + (−3) + 36                                                = −2

Step 9. Write this answer in row 2 column 2 in the (2 × 2) matrix. 