Inverse Matrices (2x2)

Inverse Matrices (2x2)

How to find the inverse matrix of a matrix (2x2): formula, proof, examples, and their solutions.

Formula

For a 2x2 matrix, the inverse matrix A^-1 is (1/(ad - bd))[d -b / -c a]. The multiplication of a matrix and its inverse matrix is the identity matrix.

For a 2×2 matrix A,
A-1 = (1/(ad - bc))[d -b / -c a].

ad - bc: determinant of the matrix

The inverse matrix (A-1) is a matrix
that satisfies
AA-1 = A-1A = I.

Proof

Inverse Matrices (2x2): Proof of the formula

To prove A-1 = (1/(ad - bc))[d -b / -c a] formula,
see if A and A-1 matrices satisfy
the definition of the inverse matrix:
AA-1 = A-1A = I.

AA-1 = I

Multiplying matrices

A-1A = I

Multiplying matrices

So if A-1 = (1/(ad - bc))[d -b / -c a],
AA-1 = A-1A = I.

So the inverse matrix formula is
A-1 = (1/(ad - bc))[d -b / -c a].

Example 1

If A = [1 2 / 3 4], find A^-1.

In A-1 = (1/(ad - bc))[d -b / -c a],
ad - bc is the determinant of the matrix.

So first find the determinant:
1⋅4 - 2⋅3 = -2.

Determinant of a matrix (2x2)

Then A-1 = (1/(-2))[4 -2 / -3 1].

Example 2

If A = [1 2 / 3 4], find A^-1.

Find the determinant:
6⋅4 - 8⋅3 = 0.

The determinant is 0.
Then the inverse matrix doesn't exist.

Here's why:

The determinant is 0.
So A-1 = (1/0)[4 -8 / -3 6].
The denominator is 0,
which doesn't make sense.

So if the determinant is 0,
then A-1 doesn't exist.