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CBSE · Class 12 · Mathematics

Invertible Matrices (Ch 3: 3.7)

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

Inverse of a matrix and how to check it

From daily life

Multiplying by 5 is undone by multiplying by 1/5. For matrices, multiplying by the inverse undoes a multiplication and brings back the identity.

Simple definition

For a square matrix A, a matrix B with AB = BA = I is the inverse of A, written A⁻¹. A matrix has at most one inverse. If a matrix has an inverse it is invertible.

Formula: A A⁻¹ = A⁻¹ A = I

Rule 1: Check an inverse — Multiply the two matrices and see if you get I.

  • Easy: [1 1 ; 0 1] [1 −1 ; 0 1] = [1 0 ; 0 1], so they are inverses of each other.
  • Medium: [1 −1 ; −1 2] [2 1 ; 1 1] = [1 0 ; 0 1].
  • Exam-type: If A² = I, then A⁻¹ = A.

Rule 2: Spot matrices without an inverse — A zero row or column kills any product.

  • Easy: The zero matrix has no inverse because O B = O, never I.
  • Medium: [1 2 ; 2 4] has no inverse (its second row is twice the first).
  • Exam-type: Only a square matrix can be invertible.
Remember

If B is the inverse of A, then A is the inverse of B.

Common mistake

Checking only AB = I; for matrices check both products (they agree for squares).

Memory clue

Inverse = the "undo" button; A times undo gives I.

Practice questions

Easy
  1. 1Is [1 −1 ; 0 1] the inverse of [1 1 ; 0 1]?
  2. 2Is [2 1 ; 1 1] the inverse of [1 −1 ; −1 2]?
  3. 3If A B = I for square matrices, what is B?
Medium
  1. 4Is the zero matrix of order 2 invertible?
  2. 5Write the inverse of [3 0 ; 0 2].
  3. 6If A² = I, what is A⁻¹?
Challenging
  1. 7If A⁻¹ = [1 −1 ; 0 1], find A.
  2. 8Can a matrix have two different inverses?
Show answers (try them yourself first)
  1. 1Yes, the product is I
  2. 2Yes, the product is I
  3. 3B = A⁻¹
  4. 4No, O B = O is never I
  5. 5[⅓ 0 ; 0 ½]
  6. 6A⁻¹ = A
  7. 7A = [1 1 ; 0 1]
  8. 8No: if B and C are inverses, B = B (A C) = (B A) C = C

Inverse of a 2 × 2 matrix

From daily life

Undoing a turn of a steering wheel needs a turn in the opposite direction by the same amount. For a 2 × 2 matrix the "opposite" is found by swapping two entries and flipping the sign of two others.

Simple definition

For A = [a b ; c d], if ad − bc ≠ 0 then A⁻¹ = 1/(ad − bc) [d −b ; −c a]. If ad − bc = 0 there is no inverse. Also (AB)⁻¹ = B⁻¹A⁻¹ and (A′)⁻¹ = (A⁻¹)′.

Formula: A⁻¹ = 1/(ad − bc) × [d −b ; −c a]

Rule 1: Find the inverse — Swap the diagonal entries, negate the others, divide by ad − bc.

  • Easy: A = [3 1 ; 5 2]: ad − bc = 1, so A⁻¹ = [2 −1 ; −5 3].
  • Medium: A = [2 4 ; 1 3]: ad − bc = 2, so A⁻¹ = ½ [3 −4 ; −1 2].
  • Exam-type: A = [4 2 ; 6 3]: ad − bc = 0, so no inverse exists.

Rule 2: Use the properties — Reverse the order for a product.

  • Easy: The inverse of I is I.
  • Medium: The inverse of the diagonal matrix [2 0 ; 0 5] is [½ 0 ; 0 ⅕].
  • Exam-type: For A = [1 2 ; 0 1], A⁻¹ = [1 −2 ; 0 1], (A⁻¹)′ = [1 0 ; −2 1] = (A′)⁻¹.
Remember

The number ad − bc decides everything: zero means no inverse.

Common mistake

Forgetting to divide by ad − bc after swapping and negating.

Memory clue

"Swap the diagonal, negate the others, divide by the determinant."

Practice questions

Easy
  1. 1Find the inverse of [1 2 ; 3 7].
  2. 2Find the inverse of [5 2 ; 3 1].
  3. 3Find the inverse of [3 2 ; 4 3].
Medium
  1. 4Does [2 6 ; 1 3] have an inverse?
  2. 5Find the inverse of [2 4 ; 1 3].
  3. 6For which k is [k 2 ; 3 6] not invertible?
Challenging
  1. 7For A = [1 2 ; 3 5] and B = [2 1 ; 1 1], find (AB)⁻¹ as B⁻¹A⁻¹.
  2. 8Find the inverse of [2 1 ; 5 3] and check that A A⁻¹ = I.
Show answers (try them yourself first)
  1. 1ad − bc = 1, so [7 −2 ; −3 1]
  2. 2ad − bc = −1, so [−1 2 ; 3 −5]
  3. 3ad − bc = 1, so [3 −2 ; −4 3]
  4. 4No, ad − bc = 6 − 6 = 0
  5. 5½ [3 −4 ; −1 2]
  6. 66k − 6 = 0 gives k = 1
  7. 7A⁻¹ = [−5 2 ; 3 −1] and B⁻¹ = [1 −1 ; −1 2], so (AB)⁻¹ = [−8 3 ; 11 −4]
  8. 8A⁻¹ = [3 −1 ; −5 2] and the product A A⁻¹ = [1 0 ; 0 1]