# Inverse Matrices (2x2)

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

## Formula

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*^{-1}*A* = *I*.

## Proof

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*^{-1}*A* = *I*.

*AA*^{-1} = *I*

Multiplying matrices

*A*^{-1}*A* = *I*

Multiplying matrices

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

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

## Example 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

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.