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Matrices and determinants: Class 9 Mathematics
Order and types of matrices, addition, subtraction and multiplication, transpose, the determinant, singular and non-singular matrices, the adjoint and inverse of a 2 by 2 matrix, and solving equations with matrices.
Key points
A matrix is a rectangular arrangement of numbers in rows and columns, written inside brackets. The numbers are the entries or elements. A matrix with m rows and n columns has order m × n (read m by n). The entry in the ith row and jth column is written a_ij.
Types of matrices:
- Row matrix: only one row, order 1 × n.
- Column matrix: only one column, order m × 1.
- Rectangular matrix: the number of rows is not equal to the number of columns.
- Square matrix: the number of rows equals the number of columns.
- Zero (null) matrix: every entry is 0.
- Diagonal matrix: a square matrix whose entries off the main diagonal are all 0.
- Scalar matrix: a diagonal matrix whose diagonal entries are all equal.
- Identity (unit) matrix I: a scalar matrix whose diagonal entries are 1.
Two matrices can be added or subtracted only if they have the same order. The entries in the same positions are added or subtracted. A matrix can be multiplied by a number (a scalar) by multiplying every entry by it.
The product AB is defined only if the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, the product has order m × p. Each entry of AB is found by multiplying the entries of a row of A by the matching entries of a column of B and adding. In general AB is not equal to BA.
The transpose of a matrix A, written Aᵗ, is found by changing its rows into columns. If a matrix equals its own transpose it is called symmetric.
A determinant is a number found from a square matrix. For the 2 × 2 matrix with rows (a, b) and (c, d), the determinant is ad − bc. A square matrix whose determinant is 0 is called singular. If the determinant is not 0 it is non-singular.
The adjoint of the 2 × 2 matrix with rows (a, b) and (c, d) is the matrix with rows (d, −b) and (−c, a). The inverse of a non-singular matrix A is its adjoint divided by its determinant. A matrix that is singular has no inverse. The product of a matrix and its inverse is the identity matrix.
A pair of linear equations can be solved with matrices. Write them as AX = B, where A holds the coefficients, X holds the unknowns and B holds the constants. If A is non-singular, then X = A⁻¹B.
Study tip: After finding an inverse, multiply it by the original matrix. If you do not get the identity matrix, there is a mistake.
Definitions and flashcards
Say the meaning first, then open the card to check.
Matrix
A rectangular arrangement of numbers in rows and columns.
Order of a matrix
The number of rows by the number of columns, for example 2 × 3.
Square matrix
A matrix with the same number of rows and columns.
Identity matrix
A square matrix with 1 on the main diagonal and 0 elsewhere.
Transpose
The matrix formed by changing the rows of a matrix into columns.
Determinant
A number found from a square matrix, ad − bc for a 2 × 2 matrix.
Singular matrix
A square matrix whose determinant is zero.
Non-singular matrix
A square matrix whose determinant is not zero.
Adjoint
For a 2 × 2 matrix, the matrix obtained by swapping the diagonal entries and changing the signs of the other two.
Inverse
The matrix that gives the identity when multiplied with the original matrix.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Order of a product | (m × n)(n × p) = m × p | Columns of A must equal rows of B. |
| Determinant of a 2 × 2 matrix | |A| = ad − bc | For the matrix with rows (a, b) and (c, d). |
| Adjoint of a 2 × 2 matrix | adj A = rows (d, −b) and (−c, a) | Swap a and d, change the signs of b and c. |
| Inverse | A⁻¹ = adj A ÷ |A| | Exists only when |A| ≠ 0. |
| Solving equations | X = A⁻¹B | For the system AX = B. |
| Product with the inverse | AA⁻¹ = A⁻¹A = I | Use it to check an inverse. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define a matrix. What is meant by the order of a matrix? (2 marks)
A matrix is a rectangular arrangement of numbers in rows and columns. The order is the number of rows by the number of columns, so a matrix with 2 rows and 3 columns has order 2 × 3.
What is a scalar matrix? How does it differ from the identity matrix? (2 marks)
A scalar matrix is a diagonal matrix whose diagonal entries are all the same number. The identity matrix is the scalar matrix in which that number is 1.
When can two matrices be multiplied? (2 marks)
The product AB exists only when the number of columns of A equals the number of rows of B.
Is matrix multiplication commutative? Explain. (3 marks)
No. In general AB is not equal to BA. In some cases BA is not even defined when AB is defined, and when both exist they may be different matrices.
What is a singular matrix? Does it have an inverse? (2 marks)
A singular matrix is a square matrix whose determinant is zero. It has no inverse because the inverse needs division by the determinant.
Describe the steps for finding the inverse of a 2 × 2 matrix. (3 marks)
Find the determinant ad − bc. If it is not zero, find the adjoint by swapping a and d and changing the signs of b and c. Divide every entry of the adjoint by the determinant.
Explain how to solve a pair of linear equations by the matrix method. (5 marks)
Write the equations in the form AX = B, where A is the matrix of coefficients, X is the column of unknowns and B is the column of constants. Find the determinant of A and check it is not zero. Find the inverse of A. Multiply both sides by the inverse to get X = A⁻¹B. Multiply the matrices and read off the values of the unknowns, then check them in the original equations.
Numericals with worked steps
Find the sum A + B where A has rows (1, 2) and (3, 4), and B has rows (5, 6) and (7, 8).
- Both matrices are 2 × 2, so they can be added.
- Add the entries in the same positions: 1 + 5 = 6, 2 + 6 = 8, 3 + 7 = 10, 4 + 8 = 12.
Answer: The matrix with rows (6, 8) and (10, 12)
Find the product AB where A has rows (1, 2) and (3, 4), and B has rows (2, 0) and (1, 3).
- A is 2 × 2 and B is 2 × 2, so AB is 2 × 2.
- Row 1 of AB: (1×2 + 2×1, 1×0 + 2×3) = (4, 6).
- Row 2 of AB: (3×2 + 4×1, 3×0 + 4×3) = (10, 12).
Answer: The matrix with rows (4, 6) and (10, 12)
Find the determinant of the matrix with rows (3, 2) and (1, 1).
- Determinant = ad − bc with a = 3, b = 2, c = 1, d = 1.
- = (3)(1) − (2)(1) = 3 − 2 = 1.
Answer: 1
Find the inverse of the matrix A with rows (3, 2) and (1, 1).
- |A| = 3(1) − 2(1) = 1, which is not zero, so the inverse exists.
- adj A has rows (1, −2) and (−1, 3).
- A⁻¹ = adj A ÷ 1.
Answer: A⁻¹ has rows (1, −2) and (−1, 3)
Solve x + y = 3 and x − y = 1 by the matrix method.
- Write AX = B with A having rows (1, 1) and (1, −1), X = (x, y) and B = (3, 1).
- |A| = (1)(−1) − (1)(1) = −2.
- adj A has rows (−1, −1) and (−1, 1), so A⁻¹ has rows (1/2, 1/2) and (1/2, −1/2).
- X = A⁻¹B: x = (1/2)(3) + (1/2)(1) = 2 and y = (1/2)(3) − (1/2)(1) = 1.
- Check: 2 + 1 = 3 and 2 − 1 = 1.
Answer: x = 2 and y = 1
Common mistakes
- Adding or subtracting matrices of different orders.
- Multiplying entries position by position instead of row by column.
- Forgetting that AB and BA can differ.
- Changing the wrong signs when writing the adjoint. Swap the diagonal entries, but only change the signs of the other two.
- Finding an inverse without first checking that the determinant is not zero.