Matrices
How to Find the inverse of a 2×2 matrix
Use the inverse of a 2×2 matrix to solve a pair of simultaneous equations written in matrix form.
Before you start
- Finding the determinant ad − bc of a 2×2 matrix
- Multiplying a 2×2 matrix by a number and by a column
- Working with fractions
When to use it
Use the inverse of a 2×2 matrix to solve a pair of simultaneous equations written in matrix form.
The steps
- For the matrix [[a, b], [c, d]], compute the determinant ad − bc.
- If the determinant is zero, stop, there is no inverse.
- Swap a and d, and change the signs of b and c.
- Multiply the new matrix by 1 divided by the determinant.
- To solve equations, multiply this inverse by the column of constants.
Worked example
Find the inverse of M = (2 1; 3 2).
- For (a b; c d) with a = 2, b = 1, c = 3, d = 2, the determinant is ad − bc = 2×2 − 1×3 = 4 − 3 = 1.
- The determinant is 1, which is not zero, so the inverse exists.
- Swap a and d, and change the signs of b and c: (2 −1; −3 2).
- Multiply by 1 ÷ determinant = 1 ÷ 1 = 1, which leaves it unchanged.
- So the inverse is (2 −1; −3 2).
A second example, with a twist
The determinant is 5, not 1, so the inverse carries a fraction, and it is used to solve a pair of simultaneous equations. Use a matrix inverse to solve 2x + y = 5 and x + 3y = 10.
- Write in matrix form (2 1; 1 3)(x; y) = (5; 10), so a = 2, b = 1, c = 1, d = 3; determinant = 2×3 − 1×1 = 6 − 1 = 5.
- The determinant is 5, not zero, so the inverse exists.
- Swap a and d, and change the signs of b and c: (3 −1; −1 2).
- Multiply by 1 ÷ 5: the inverse is (1/5)(3 −1; −1 2).
- Multiply the inverse by the column (5; 10): (1/5)(3×5 + (−1)×10; (−1)×5 + 2×10) = (1/5)(5; 15) = (1; 3), so x = 1 and y = 3.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
What if the determinant is zero?
Then the matrix has no inverse, it is called singular. Dividing by zero is impossible, so you cannot form the inverse.
For simultaneous equations this usually means the two equations have no unique solution: they may have none, or infinitely many. Stop as soon as you see a zero determinant.
Which numbers do I swap and which do I change signs?
Swap the two numbers on the main diagonal, a and d, the top-left and bottom-right. Then change the signs of the other two, b and c, without moving them.
A handy check: multiply your inverse by the original matrix and you should get the identity matrix (1 0; 0 1).
Why does the inverse solve the equations?
Writing the equations as M times the column (x; y) equals the constants lets you multiply both sides by M⁻¹. Because M⁻¹ times M is the identity, the left side becomes just (x; y).
So the inverse times the constant column gives the values of x and y directly.