Inverse of [2, 1; 1, 1] Step by Step | Inverse of a 2 × 2 matrix

Step by step

The question

Question: find the inverse of A.

A = [2, 1; 1, 1]

The inverse of A is written A⁻¹.

A times its inverse gives I: 1s down the diagonal, 0s elsewhere.

Times I changes nothing, like times 1.

A × A⁻¹ = I, I = [1, 0; 0, 1]

1 Find the determinant

First we need D, the determinant of A.

D = |2, 1; 1, 1|

So D = 2 × 1 − 1 × 1.

D = 2 × 1 − 1 × 1

2 times 1 is 2.

1 times 1 is 1.

D = 2 − 1

2 minus 1 is 1.

So D = 1.

D = 1

2 Swap and change signs

The top left and bottom right swap places: 2 and 1.

The other two stay where they are, but change their signs: 1 becomes −1, 1 becomes −1.

Then we divide by D.

Dividing by 1 is the same as times 1/1.

A⁻¹ = 11 × [1, −1; −1, 2]

Careful!

Only the other two change sign.

2 and 1 just swap places.

Careful: 2 and 1 swap, keep their signs

3 Divide by the determinant

Dividing by 1 changes nothing.

So the answer is rows 1, −1 and −1, 2.

A⁻¹ = [1, −1; −1, 2]

4 Check

Let's check: A times the swapped matrix.

It must give 1s down the diagonal, 0s elsewhere.

= [2, 1; 1, 1] × [1, −1; −1, 2]

= [2 × 1 + 1 × (−1), 2 × (−1) + 1 × 2; 1 × 1 + 1 × (−1), 1 × (−1) + 1 × 2]

We work out every times.

= [2 + (−1), −2 + 2; 1 + (−1), −1 + 2]

Then we add: 1 down the diagonal, 0 everywhere else.

= [1, 0; 0, 1]

Divided by 1, that is I.

So A times our answer is I: the answer is right!

A × A⁻¹ = I

Let's look back.

First: find the determinant.

Then: swap and change signs.

Then: divide by the determinant.

Last: check.

Answer