medium difficulty8 min

Adjoint and Inverse of a Matrix

The adjoint of a square matrix A is the transpose of its cofactor matrix. If |A| ≠ 0, then A inverse exists and A⁻¹ = adj(A)/|A|.

In one line: Adjoint is the flipped cofactor matrix; inverse is adjoint divided by the determinant.

Practice This Concept

Understand it

The inverse formula using determinants is built on the identity A adj(A) = adj(A) A = |A|I.
This means adj(A) behaves like the numerator of the inverse, but division by |A| is allowed only when |A| is non-zero.
In an exam solution, first test |A|.
If |A| = 0, A is singular and no inverse exists; calculating cofactors after that is unnecessary for inverse.
If |A| ≠ 0, find every cofactor, arrange them in the cofactor matrix, transpose that matrix to get adj(A), and divide each entry of adj(A) by |A|.

Example

For A = [[1,2],[3,4]], |A| = 1×4 - 2×3 = -2. adj(A) = [[4,-2],[-3,1]]. Hence A⁻¹ = (-1/2)[[4,-2],[-3,1]] = [[-2,1],[3/2,-1/2]].

Formula and rule

adj(A) = transpose of the cofactor matrix.

A adj(A) = adj(A) A = |A|I.

If |A| ≠ 0, A⁻¹ = adj(A)/|A|.

If |A| = 0, A is singular and non-invertible.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find inverse of A = [[2,1],[5,3]]. |A| = 2×3 - 1×5 = 1. adj(A) = [[3,-1],[-5,2]].

1Therefore A⁻¹ = [[3,-1],[-5,2]].
2Check: A A⁻¹ = [[1,0],[0,1]], so the inverse is correct.

Method you can reuse

1Confirm the matrix is square.
2Find |A|.
3If it is zero, state no inverse.
4If non-zero, find all minors and cofactors.
5Arrange cofactors into the cofactor matrix.
6Transpose to get adj(A).
7Divide every entry by |A| and verify if needed.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What condition must hold for A⁻¹ to exist?

Think first, then show answer

A must be square and |A| must be non-zero.

Common wrong answer

Dividing the original matrix by |A| instead of dividing adj(A) by |A|.

Don't mix this up

Using the cofactor matrix directly as adj(A) without transposing it.

Exam tip

Before finding cofactors, check |A|. If |A| = 0, stop and state that inverse does not exist.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to find A inverse, prove a matrix is singular, verify A adj(A) = |A|I, or use inverse in a system of equations.

1
Enough for 1 mark
The adjoint of a square matrix A is the transpose of its cofactor matrix. If |A| ≠ 0, then A inverse exists and A⁻¹ = adj(A)/|A|.
2
Add for 2 marks
Example: For A = [[1,2],[3,4]], |A| = 1×4 - 2×3 = -2. adj(A) = [[4,-2],[-3,1]]. Hence A⁻¹ = (-1/2)[[4,-2],[-3,1]] = [[-2,1],[3/2,-1/2]].
3
Add for 3 marks
The inverse formula using determinants is built on the identity A adj(A) = adj(A) A = |A|I.

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