Inverse of [4, 7; 2, 4]

Step by step

The question

Question: find the inverse of A.

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.

1 Find the determinant

First we need D, the determinant of A.

So D = 4 × 4 − 2 × 7.

4 times 4 is 16.

2 times 7 is 14.

16 minus 14 is 2.

So D = 2.

2 Swap and change signs

The top left and bottom right swap places: 4 and 4.

The other two stay where they are, but change their signs: 7 becomes −7, 2 becomes −2.

Then we divide by D.

Dividing by 2 is the same as times 1/2.

Careful!

Only the other two change sign.

4 and 4 just swap places.

3 Divide by the determinant

Every entry over 2.

Then we make each one simpler.

The answer: rows 2, −7/2 and −1, 2.

4 Check

Let's check: A times the swapped matrix.

It must give 2s down the diagonal, 0s elsewhere.

We work out every times.

Then we add: 2 down the diagonal, 0 everywhere else.

Divided by 2, that is I.

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

Let's look back.

First: find the determinant.

Then: swap and change signs.

Then: divide by the determinant.

Last: check.