JENECONKMaths Universe

Inverse of a 2 × 2 Matrix

Ages 14–15

Difficulty adapts to you

Inverse of a 2 × 2 Matrix

The inverse A⁻¹ reverses the effect of A. It exists only when det(A) is non-zero. For A = [[a, b], [c, d]], A⁻¹ = 1/(ad − bc) [[d, −b], [−c, a]]. First calculate the determinant, then swap a and d, change the signs of b and c, and multiply the whole matrix by the reciprocal of the determinant. Verify an answer by checking AA⁻¹ = I.

Your goals

  • ✓You can find and verify the inverse of a non-singular 2 × 2 matrix.
  • ✓You can use inverse, adjugate, reciprocal correctly.
  • ✓You can calculate the determinant first and verify A A⁻¹ = I.

Words to know

inverse:
A matrix that multiplies with A to give the identity matrix.
adjugate:
The matrix formed by swapping the diagonal and negating the off-diagonal entries.
reciprocal:
One divided by a non-zero number, such as 1/det(A).

Examples

  • 1For A = 3121, det(A) = 1, so A⁻¹ = 1−1−23.
  • 2For B = 2111, det(B) = 1, so B⁻¹ = 1−1−12.
  • 3For C = 4263, det(C) = 12 − 12 = 0, so C has no inverse.
  • 4Do not change only the diagonal entries: the off-diagonal entries must also change sign before multiplying by 1/det(A).
  • 5If AA⁻¹ gives 1001, the calculation is consistent; any other result signals an error.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Adjugate formula

Best for: Finding a 2 × 2 inverse
  1. 1.Find ad − bc.
  2. 2.Swap a and d and change the signs of b and c.
  3. 3.Multiply by 1/det(A).

Example: [[a, b], [c, d]]⁻¹ = 1/(ad−bc)[[d, -b], [-c, a]].

Method 2

Identity check

Best for: Verifying an inverse
  1. 1.Multiply A by your proposed inverse.
  2. 2.Check all four entries.
  3. 3.Accept the result only if it is I.

Example: A A⁻¹ should equal [[1, 0], [0, 1]].

Check these

  • • Do not divide by a zero determinant or change the wrong signs in the inverse formula.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own inverse of a 2 × 2 matrix example, solve it, and explain how you checked it.

Professor Wise Owl

Professor Wise Owl

Wisdom & Understanding

“Mistakes are proof that you are trying. Keep going!”