Matrices and Determinant – SS3 Further Mathematics Lesson Note (First Term)

A matrix is a rectangular array of numbers, quantities or expressions arranged in rows and columns and enclosed in brackets.

Behavioural objectives

By the end of the lesson, students should be able to:

1. Define a matrix and identify one.
2. State the order and notation of a matrix.
3. Name and define the main types of matrices.
4. Find the transpose of a matrix.
5. Add and subtract matrices of the same order.
6. Find the determinant of a 2 by 2 matrix.

Instructional resources

• Matrix charts: show rows, columns and the order of a matrix.
• Addition and subtraction charts: show matching entries being added or subtracted.

What is a matrix?

A matrix is a rectangular array of numbers, quantities or expressions.

• Element (entry): each number or expression inside the matrix.
• Row: a horizontal line of elements.
• Column: a vertical line of elements.
• Notation: a capital letter names a matrix, for example A, B or X.
• Brackets: round brackets ( ) or square brackets [ ] enclose the elements.

Example: class composition by house and gender

The table shows the students in a class by house and gender.

Calling the entries X, we get the following matrices.

(i) X =

(ii) A matrix with two rows and three columns:

(iii) A matrix with one row: (6 4 2 1)

• Teacher: probes what students know, then leads them to define a matrix.
• Students: give more examples of matrices from everyday data, such as scores or prices.

What is the order of a matrix?

The order of a matrix is the number of rows by the number of columns, written m × n.

• Rule: always state rows first, then columns.
• Matrix (i) above: 4 rows and 2 columns, so the order is 4 × 2.
• Matrix (ii) above: 2 rows and 3 columns, so the order is 2 × 3.
• Matrix (iii) above: 1 row and 4 columns, so the order is 1 × 4.
• Element notation: a₂₃ means the element in row 2, column 3.

What are the types of matrices?

There are six common types of matrices: zero, row, column, square, diagonal and identity.

Types and examples

• Null (zero) matrix: every element is zero, for example the 2 × 2 matrix with rows (0 0) and (0 0).
• Row matrix: a single row of elements, for example (2 3 4), of order 1 × 3.
• Column matrix: a single column of elements, for example the entries 5, 1 and 7 written one below another, of order 3 × 1.
• Square matrix: the number of rows equals the number of columns, for example 2 × 2 or 3 × 3.
• Diagonal matrix: a square matrix whose elements are all zero except those on the main diagonal (top left to bottom right).
• Identity (unit) matrix: a diagonal matrix with 1 on every main diagonal position, written I.

What is the transpose of a matrix?

The transpose of a matrix is the matrix formed by interchanging its rows and columns, written A^T.

• Order: if A is m × n, then A^T is n × m.
• Example: matrix (ii) above has order 2 × 3, so its transpose has order 3 × 2.

Transpose of matrix (ii):

• Students: write the transpose of matrices (i) and (iii) above and state the order of each.

How do we add and subtract matrices?

To add or subtract matrices, add or subtract the elements in the same position.

Rules

• Same order: only matrices of the same order can be added or subtracted.
• Result: the answer has the same order as the matrices used.
• Addition: A + B = B + A, so addition is commutative.
• Subtraction: A – B is not equal to B – A in general, so subtraction is not commutative.

Worked example

Given A and B below, find A + B, B + A and A – B.

A =

B =

1. Check that A and B are both 2 × 3. They are, so the sums exist.
2. Add each pair of matching elements: 6 + (-1) = 5, 3 + 3 = 6, 1 + 4 = 5, 2 + 5 = 7, 3 + 0 = 3, 4 + 2 = 6.
3. A + B has rows (5 6 5) and (7 3 6).
4. B + A gives the same result, which shows addition is commutative.
5. Subtract matching elements for A – B: 6 – (-1) = 7, 3 – 3 = 0, 1 – 4 = -3, 2 – 5 = -3, 3 – 0 = 3, 4 – 2 = 2.
6. A – B has rows (7 0 -3) and (-3 3 2).

• Students: find B – A and compare it with A – B. Each element has the opposite sign.

What is the determinant of a 2 by 2 matrix?

The determinant of a square matrix is a single number calculated from its elements.

Formula

• Matrix: for a matrix with rows (a b) and (c d), the determinant is written |A|.
• Rule: |A| = ad – bc, which is the product of the main diagonal minus the product of the other diagonal.
• Singular matrix: a matrix whose determinant is 0.
• Non-singular matrix: a matrix whose determinant is not 0.

Worked examples

1. Matrix with rows (3 2) and (1 4): |A| = (3 × 4) – (2 × 1) = 12 – 2 = 10. It is non-singular.
2. Matrix with rows (2 4) and (1 2): |A| = (2 × 2) – (4 × 1) = 4 – 4 = 0. It is singular.

Evaluation

1. Define a matrix and state the order of a matrix with 3 rows and 2 columns.
2. List five types of matrices, giving an example of each.
3. Is addition of matrices commutative? Is subtraction commutative? Give a reason.
4. Given X and Y below, find X + Y.
5. Find the determinant of Y and state whether it is singular.

• Students: solve the problems while the teacher goes round to check their working.

Assignment

Given the matrices P and Q below, find:

1. P + Q
2. P – Q
3. Q – P
4. The determinant of P

References

• New Concept Mathematics
• Multipurpose Mathematics