Matrices and determinants: Class 11 Mathematics Notes, Short Questions and Numericals

Matrices and their types, operations, transpose, determinants of order 2 and 3, minors and cofactors, adjoint and inverse, and solving linear equations by the matrix method and Cramer's rule.

Key points

A matrix is a rectangular array of numbers arranged in rows and columns and enclosed in brackets. A matrix with m rows and n columns has order m × n, and the number in row i and column j is written aᵢⱼ. In this text a matrix is written row by row, with rows separated by semicolons, so [1 2; 3 4] has rows (1, 2) and (3, 4). Two matrices are equal when they have the same order and equal entries in the same places.

Types of matrices:

Row matrix: one row. Column matrix: one column.
Square matrix: the number of rows equals the number of columns.
Zero matrix: every entry is 0.
Diagonal matrix: a square matrix with all entries off the main diagonal equal to 0.
Scalar matrix: a diagonal matrix with equal diagonal entries.
Identity matrix I: a diagonal matrix with every diagonal entry 1.
Symmetric matrix: A equals its transpose. Skew symmetric matrix: A equals minus its transpose, so its diagonal entries are 0.

Matrices of the same order can be added or subtracted by adding or subtracting the entries in the same places. Multiplying a matrix by a scalar k multiplies every entry by k. Addition is commutative and associative, and the zero matrix acts as the additive identity.

Two matrices A and B can be multiplied 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, then AB is m × p. The entry in row i and column j of AB is found by multiplying the entries of row i of A by the matching entries of column j of B and adding. In general AB ≠ BA, and AB can be zero even when neither A nor B is zero. Multiplication is associative and distributes over addition, and AI = IA = A.

The transpose Aᵀ of a matrix is found by turning its rows into columns. If A is m × n then Aᵀ is n × m. The rules are (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (kA)ᵀ = kAᵀ and (AB)ᵀ = BᵀAᵀ. The order reverses for a product.

The determinant is a number linked to a square matrix A and is written |A| or det A. For a 2 × 2 matrix [a b; c d], |A| = ad − bc. For a 3 × 3 matrix we expand along a row or column. The minor Mᵢⱼ of an entry is the determinant left after deleting its row and column. The cofactor is Cᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ. The determinant is the sum of each entry of a chosen row times its cofactor.

Properties of determinants:

Interchanging two rows changes the sign of the determinant.
If two rows (or columns) are identical or proportional, the determinant is 0.
A common factor of a row can be taken out of the determinant.
Adding a multiple of one row to another row does not change the value.
|AB| = |A||B| and |Aᵀ| = |A|.

A square matrix is singular if |A| = 0 and non singular if |A| ≠ 0. The adjoint of A, written adj A, is the transpose of the matrix of cofactors. It satisfies A(adj A) = (adj A)A = |A| I. If A is non singular, its inverse is A⁻¹ = (adj A)/|A|, and AA⁻¹ = A⁻¹A = I. A singular matrix has no inverse. For a 2 × 2 matrix [a b; c d], the adjoint is [d −b; −c a].

The inverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹. Also (A⁻¹)⁻¹ = A and (Aᵀ)⁻¹ = (A⁻¹)ᵀ.

A system of linear equations AX = B, with A non singular, has the unique solution X = A⁻¹B. Cramer's rule gives the same result with determinants. For two equations in x and y, x = Dₓ/D and y = Dᵧ/D, where D is the determinant of the coefficients and Dₓ is D with the x column replaced by the constants. If D = 0 the system has either no solution or infinitely many solutions. A system with all constants zero is homogeneous, and it always has the zero solution.

Study tip: Always check the order of matrices before multiplying. For a 3 × 3 determinant, expand along the row or column with the most zeros, and write each term on its own line with its sign. After finding an inverse, multiply it back by the original matrix to confirm the identity.

Definitions and flashcards

Say the meaning first, then open the card to check.

Order of a matrix

The number of rows by the number of columns, written m × n.

Identity matrix

A square matrix with 1 on the main diagonal and 0 elsewhere.

Transpose

The matrix obtained by interchanging the rows and columns of a matrix.

Minor

The determinant left when the row and column of an entry are deleted.

Cofactor

The minor of an entry multiplied by (−1)ⁱ⁺ʲ.

Singular matrix

A square matrix whose determinant is 0.

Adjoint

The transpose of the matrix of cofactors.

Inverse

The matrix A⁻¹ with AA⁻¹ = A⁻¹A = I.

Formula sheet

Name	Formula	What it means
Determinant of order 2	\|a b; c d\| = ad − bc	Cross product of the diagonals.
Cofactor	Cᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ	Used for expansion and the adjoint.
Inverse	A⁻¹ = (adj A)/\|A\|	Needs \|A\| ≠ 0.
Product rule	\|AB\| = \|A\|\|B\|	For square matrices of the same order.
Transpose of a product	(AB)ᵀ = BᵀAᵀ	The order reverses.
Cramer's rule	x = Dₓ/D, y = Dᵧ/D	For two equations with D ≠ 0.

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.

When can two matrices be multiplied? What is the order of the product? (2 marks)

A and B can be multiplied when 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 AB has order m × p.

Define a singular matrix. Does it have an inverse? (2 marks)

A square matrix with determinant 0 is singular. It has no inverse, because the inverse formula A⁻¹ = (adj A)/|A| needs division by |A|.

Is matrix multiplication commutative? Show with an example. (3 marks)

No. Take A = [1 2; 0 1] and B = [1 0; 3 1]. Then AB = [7 2; 3 1] and BA = [1 2; 3 7]. These are different, so AB ≠ BA. Matrix multiplication is therefore not commutative in general.

State any three properties of determinants. (3 marks)

First, if two rows or columns are interchanged the sign of the determinant changes. Second, if two rows or columns are identical, the determinant is 0. Third, adding a multiple of one row to another row leaves the value unchanged. A common factor of a row can also be taken outside.

Explain how to find the inverse of a 2 × 2 matrix [a b; c d]. (5 marks)

First find the determinant D = ad − bc. If D = 0 the matrix is singular and there is no inverse. If D ≠ 0, form the adjoint by swapping a and d and changing the signs of b and c, which gives [d −b; −c a]. Then divide every entry by D. So A⁻¹ = (1/D)[d −b; −c a]. Check by confirming that AA⁻¹ = I.

Explain how to solve two linear equations by the matrix method. (5 marks)

Write the system as AX = B, where A holds the coefficients, X = [x; y] and B holds the constants. Find |A|. If |A| ≠ 0, find A⁻¹ = (adj A)/|A|. Multiply both sides on the left by A⁻¹ to get X = A⁻¹B. Read x and y from the product. If |A| = 0 the method fails, and the system must be checked for no solution or many solutions.

Numericals with worked steps

Let A = [1 2; 3 4] and B = [0 1; 5 2]. Find AB and BA.

AB row 1: (1)(0) + (2)(5) = 10 and (1)(1) + (2)(2) = 5.
AB row 2: (3)(0) + (4)(5) = 20 and (3)(1) + (4)(2) = 11.
BA row 1: (0)(1) + (1)(3) = 3 and (0)(2) + (1)(4) = 4.
BA row 2: (5)(1) + (2)(3) = 11 and (5)(2) + (2)(4) = 18.

Answer: AB = [10 5; 20 11] and BA = [3 4; 11 18], so AB ≠ BA.

Find the inverse of A = [3 1; 5 2] and verify it.

|A| = (3)(2) − (1)(5) = 6 − 5 = 1, which is not 0.
adj A = [2 −1; −5 3].
A⁻¹ = (1/1)[2 −1; −5 3] = [2 −1; −5 3].
Check row 1 of A with column 1 of A⁻¹: (3)(2) + (1)(−5) = 1.
Row 1 with column 2: (3)(−1) + (1)(3) = 0. The rest follows in the same way, giving I.

Answer: A⁻¹ = [2 −1; −5 3].

Evaluate the determinant with rows (1, 2, 3), (0, 1, 4) and (5, 6, 0).

Expand along row 1: 1·(1·0 − 4·6) − 2·(0·0 − 4·5) + 3·(0·6 − 1·5).
First term: 1·(0 − 24) = −24.
Second term: −2·(0 − 20) = +40.
Third term: 3·(0 − 5) = −15.
Sum: −24 + 40 − 15 = 1.

Answer: The determinant is 1.

Solve 2x + y = 5 and x − y = 1 by Cramer's rule.

D = (2)(−1) − (1)(1) = −3.
Dₓ = (5)(−1) − (1)(1) = −6, so x = −6/−3 = 2.
Dᵧ = (2)(1) − (5)(1) = −3, so y = −3/−3 = 1.
Check: 2(2) + 1 = 5 and 2 − 1 = 1.

Answer: x = 2 and y = 1.

Common mistakes

Multiplying matrices entry by entry. The rule is row times column and add.
Assuming AB = BA. Order matters, and (AB)ᵀ = BᵀAᵀ, not AᵀBᵀ.
Forgetting the sign pattern (+ − + / − + − / + − +) when expanding a 3 × 3 determinant.
Using the formula A⁻¹ = (adj A)/|A| without checking that |A| is not zero.
Writing the adjoint without taking the transpose of the cofactor matrix.

Practise this chapter

Practice questions and timed test Printable worksheet