Matrices

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 8 Matrices

Lesson objectives

By the end of this topic, you should be able to:

• Define a matrix, state its order and identify a square matrix.
• Add, subtract and multiply matrices.
• Find the determinant and inverse of a 2 × 2 matrix.
• Use matrices to solve simultaneous equations.

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns.

a) Order

The order is rows × columns, always in that order.

A matrix with 2 rows and 3 columns has order 2 × 3.

A square matrix has equal rows and columns.

A row matrix has one row; a column matrix has one column.

b) Adding and subtracting

Matrices can be added only if they have the same order.

Add corresponding elements, position by position.

Subtraction works the same way.

c) Multiplying by a scalar

Multiply every element by the scalar.

So 3 times (2 5 / 1 4) gives (6 15 / 3 12).

d) Multiplying two matrices

Two matrices can be multiplied only if the columns of the first equal the rows of the second.

An m × n matrix times an n × p matrix gives an m × p matrix.

Each element comes from a row of the first paired with a column of the second. Multiply corresponding entries and add.

Matrix multiplication is not commutative. AB and BA are usually different, and one may not even exist. This is the main way matrices differ from ordinary numbers.

e) The identity matrix

The 2 × 2 identity is (1 0 / 0 1).

Multiplying any matrix by the identity leaves it unchanged, exactly as multiplying a number by 1 does.

f) Determinant and inverse

For a 2 × 2 matrix (a b / c d), the determinant is ad − bc.

The inverse is (1/det) × (d −b / −c a).

So: swap the leading diagonal, negate the other diagonal, and divide by the determinant.

If the determinant is zero, no inverse exists. The matrix is singular.

Check your inverse by multiplying it back. The product must be the identity.

g) Solving simultaneous equations

Write the equations as a matrix equation AX = B. Here A holds the coefficients, X the unknowns and B the constants.

Then X = A⁻¹B.

Take 3x + 5y = 11 and x + 4y = 8. Here A is (3 5 / 1 4), with determinant 7.

Multiplying A⁻¹ by B gives x = 4/7 and y = 13/7. Check both equations to confirm.

If the determinant is zero the equations have no unique solution. They are either inconsistent or the same line twice.

h) Where this is used

Solving systems of equations. Transformations in geometry. Network and route analysis. Computer graphics.

Words to know

• Matrix -- a rectangular array of numbers in rows and columns.
• Order -- the size of a matrix, written rows × columns.
• Identity matrix -- the square matrix with 1s on the leading diagonal and 0s elsewhere.
• Determinant -- for a 2 × 2 matrix, the number ad − bc.
• Singular matrix -- one with determinant zero, having no inverse.

:::checkpoint Check yourself

1. State the order of a matrix with 3 rows and 2 columns.
2. Find the determinant of (4 3 / 2 5).
3. Find the inverse of that matrix.
4. Why is matrix multiplication not commutative? :::

Bridge to practice

The exercises begin with order and the operations, move through multiplication and non-commutativity, and finish with determinants, inverses and simultaneous equations. Before multiplying any pair of matrices, write their orders side by side and check that the inner numbers match.