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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|.

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Student-friendly explanation

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|.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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]].

  4. 4

    Avoid this incomplete answer

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

Definition

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|.

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]].

Rule to remember

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.

Memory hook

Adjoint is the flipped cofactor matrix; inverse is adjoint divided by the determinant.

Examples and method

Worked example

Find inverse of A = [[2,1],[5,3]]. |A| = 2×3 - 1×5 = 1. adj(A) = [[3,-1],[-5,2]]. Therefore A^-1 = [[3,-1],[-5,2]]. Check: A A^-1 = [[1,0],[0,1]], so the inverse is correct.

Method to apply

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

Diagram support

No diagram is needed. A cofactor-to-adjoint table can help students see the transpose step.

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

What condition must hold for A^-1 to exist?

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

Answer writing and exam use

1-mark answer

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-mark answer

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|. 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. 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-mark answer

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|. 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. Find inverse of A = [[2,1],[5,3]]. |A| = 2×3 - 1×5 = 1. adj(A) = [[3,-1],[-5,2]]. Therefore A^-1 = [[3,-1],[-5,2]]. Check: A A^-1 = [[1,0],[0,1]], so the inverse is correct. 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. Dividing the original matrix by |A| instead of dividing adj(A) by |A|.
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