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CBSE · Class 12 · Mathematics

Adjoint and Inverse of a Matrix (Ch 4: 4.5)

2 concepts · 16 practice questions with answers · Free to read

Adjoint of a matrix

From daily life

Collect every entry's cofactor into a new table and then turn it on its side. The table you get is the adjoint, the key ingredient for the inverse.

Simple definition

The adjoint adj A of a square matrix is the transpose of its matrix of cofactors. For a 2 × 2 matrix, just swap the diagonal entries and negate the other two. Always A (adj A) = (adj A) A = |A| I.

Formula: A (adj A) = |A| I, |adj A| = |A|ⁿ⁻¹ for order n

Rule 1: Find adj A — Cofactors first, then transpose.

  • Easy: adj [3 1 ; 5 2] = [2 −1 ; −5 3].
  • Medium: For A = [1 1 0 ; 0 1 1 ; 1 0 1], the cofactor matrix is [1 1 −1 ; −1 1 1 ; 1 −1 1], so adj A = [1 −1 1 ; 1 1 −1 ; −1 1 1] and |A| = 2.
  • Exam-type: A (adj A) for that A gives 2I; for example the first row times the first column is 1 + 1 + 0 = 2.

Rule 2: Use the properties — Quick results without computing.

  • Easy: For order 2, |adj A| = |A|.
  • Medium: For order 3 with |A| = 2, |adj A| = 4.
  • Exam-type: For order 3 with |A| = 3, A (adj A) = 3I.
Remember

The adjoint is the transpose of the cofactor matrix, not the cofactor matrix itself.

Common mistake

Forgetting to transpose after finding the cofactors.

Memory clue

Cofactors, then flip: adjoint.

Practice questions

Easy
  1. 1Find adj [1 2 ; 3 4].
  2. 2Find adj [5 0 ; 0 3].
  3. 3Find adj I for the 2 × 2 identity I.
Medium
  1. 4If |A| = 5 for a 3 × 3 matrix, find |adj A|.
  2. 5For A = [2 3 ; 1 4] verify that A (adj A) = |A| I.
  3. 6Find adj A for A = [2 1 0 ; 1 3 2 ; 0 1 4].
Challenging
  1. 7If A is 2 × 2 with |A| = 4, find |adj A|.
  2. 8If A is 3 × 3 with |A| = −2, find |A (adj A)|.
Show answers (try them yourself first)
  1. 1[4 −2 ; −3 1]
  2. 2[3 0 ; 0 5]
  3. 3I itself, [1 0 ; 0 1]
  4. 45² = 25
  5. 5|A| = 5, adj A = [4 −3 ; −1 2] and the product is [5 0 ; 0 5] = 5I
  6. 6[10 −4 2 ; −4 8 −4 ; 1 −2 5] (with |A| = 16)
  7. 74
  8. 8A (adj A) = −2 I, and |−2 I| = (−2)³ = −8

Inverse using the adjoint

From daily life

To undo a scrambling you need the right "undo" table. The adjoint divided by the determinant is exactly that table.

Simple definition

A square matrix A has an inverse exactly when |A| ≠ 0 (it is non-singular); then A⁻¹ = adj A / |A|. If |A| = 0 the matrix is singular and has no inverse.

Formula: A⁻¹ = (1 / |A|) adj A

Rule 1: Find the inverse — Compute |A|, then adj A, then divide.

  • Easy: [2 1 ; 5 3]: |A| = 1 and adj A = [3 −1 ; −5 2], so A⁻¹ = [3 −1 ; −5 2].
  • Medium: [1 1 0 ; 0 1 1 ; 1 0 1]: A⁻¹ = ½ [1 −1 1 ; 1 1 −1 ; −1 1 1].
  • Exam-type: [1 2 3 ; 2 4 6 ; 1 0 1] is singular because row 2 is twice row 1.

Rule 2: Singular or not? — Look at the determinant.

  • Easy: [2 4 ; 1 2] has determinant 0, so it is singular.
  • Medium: [1 2 ; 3 5] has determinant −1, so it is non-singular.
  • Exam-type: [k 1 ; 4 2] is singular when 2k − 4 = 0, i.e. k = 2.
Remember

|A| = 0 means no inverse; |A| ≠ 0 means exactly one inverse.

Common mistake

Dividing by |A| without checking that it is not zero.

Memory clue

Inverse = adjoint over determinant.

Practice questions

Easy
  1. 1Find the inverse of [3 4 ; 2 3].
  2. 2Find the inverse of [4 2 ; 3 2].
  3. 3Is [6 3 ; 4 2] invertible?
Medium
  1. 4Find the inverse of [1 2 ; 3 5].
  2. 5Find k so that [k 4 ; 3 6] is singular.
  3. 6If A⁻¹ = [1 −1 ; −1 2], find A.
Challenging
  1. 7For a 3 × 3 matrix with |A| = 3, find |A⁻¹| and |adj A|.
  2. 8For A = [2 1 0 ; 1 3 2 ; 0 1 4], find the entry in row 1, column 1 of A⁻¹.
Show answers (try them yourself first)
  1. 1|A| = 1, so [3 −4 ; −2 3]
  2. 2|A| = 2, so ½ [2 −2 ; −3 4]
  3. 3No, the determinant is 12 − 12 = 0
  4. 4|A| = −1, so [−5 2 ; 3 −1]
  5. 56k − 12 = 0, so k = 2
  6. 6A is the inverse of A⁻¹: |A⁻¹| = 1, so A = [2 1 ; 1 1]
  7. 7|A⁻¹| = 1/3 and |adj A| = 3² = 9
  8. 810 / 16 = 5/8