The inverse of a 2 × 2 matrix undoes the matrix: multiplying the two gives the identity matrix. For the matrix with rows (a, b) and (c, d), swap a and d, change the signs of b and c, and divide by ad − bc.
This lesson follows adding and multiplying matrices in the SPM Mathematics matrices chapter.
What are the three steps?
- Find the determinant: ad − bc. If it is 0, stop, because no inverse exists.
- Swap a and d. Change the signs of b and c. This gives the matrix with rows (d, −b) and (−c, a).
- Multiply that matrix by 1 ÷ (ad − bc).
The idea links to inverse functions: both undo an operation, and both are checked by doing the operation and its inverse together.
Worked example: a determinant of 1
Find the inverse of M, which has rows (3, 2) and (4, 3).
Step 1. Determinant = 3(3) − 2(4) = 9 − 8 = 1. It is not zero, so M has an inverse.
Step 2. Swap 3 and 3, and negate 2 and 4. This gives rows (3, −2) and (−4, 3).
Step 3. Multiply by 1 ÷ 1, which changes nothing. So M⁻¹ has rows (3, −2) and (−4, 3).
Check. M times M⁻¹: row 1 is 3(3) + 2(−4) = 1 and 3(−2) + 2(3) = 0. Row 2 is 4(3) + 3(−4) = 0 and 4(−2) + 3(3) = 1. The product has rows (1, 0) and (0, 1), the identity matrix.
A second example with a fraction
Find the inverse of N, which has rows (2, 1) and (5, 4).
Determinant = 2(4) − 1(5) = 8 − 5 = 3.
Swap and negate to get rows (4, −1) and (−5, 2). Then N⁻¹ = 1 ÷ 3 times that, which has rows (4/3, −1/3) and (−5/3, 2/3).
The mistake that costs marks
The common slip is to negate the wrong entries: changing the signs of a and d, or negating everything.
| Wrong | Right | |
|---|---|---|
| Matrix | Rows (3, 2) and (4, 3) | Rows (3, 2) and (4, 3) |
| Swapped matrix | Rows (−3, 2) and (4, −3) | Rows (3, −2) and (−4, 3) |
| Rule | Negated the leading diagonal | Swap the diagonal, negate the other two |
| Check | Skipped | Multiply to get the identity |
The check catches the mistake immediately. With the wrong matrix, the product is not the identity.
When is there no inverse?
Take the matrix with rows (2, 4) and (1, 2). The determinant is 2(2) − 4(1) = 0. This matrix has no inverse. Write that sentence, and show the working, because the question wants both.
Check yourself
Find the inverse of P, which has rows (5, 2) and (7, 3), and check your answer.
Answer
Determinant = 5(3) − 2(7) = 15 − 14 = 1.
Swap and negate: rows (3, −2) and (−7, 5). Dividing by 1 changes nothing, so P⁻¹ has rows (3, −2) and (−7, 5).
Check: row 1 of PP⁻¹ is 5(3) + 2(−7) = 1 and 5(−2) + 2(5) = 0. Row 2 is 7(3) + 3(−7) = 0 and 7(−2) + 3(5) = 1. The product is the identity.
What to study next
The inverse solves equations. Continue with solving simultaneous equations using matrices. Practise with the two-by-two matrix operations tutor and log slips in the mistake log and paper-error review.
If you want a teacher to check your signs as you work, see online one-to-one Mathematics tuition.