Det [1, 2, 0; 3, 1, 2; 0, 1, 4] Step by Step

Step by step

The question

Question: find the determinant of this matrix.

D = |1, 2, 0; 3, 1, 2; 0, 1, 4|

The determinant is one number we get from a square matrix.

We write the matrix between two straight lines, and call it D.

1 Copy the first two columns

We write the first two columns again, on the right.

Then every diagonal has three numbers.

D = |1, 2, 0; 3, 1, 2; 0, 1, 4|1, 2; 3, 1; 0, 1

2 Multiply down

Three diagonals go down to the right.

We multiply the three numbers on each one.

Down 1: 1 × 1 × 4.

1 × 1 × 4 is 4.

1 × 1 × 4 = 4

Down 2: 2 × 2 × 0.

2 × 2 × 0 is 0.

2 × 2 × 0 = 0

Down 3: 0 × 3 × 1.

0 × 3 × 1 is 0.

0 × 3 × 1 = 0

3 Multiply up

Three diagonals go up to the right.

We multiply those too.

Up 1: 0 × 1 × 0.

0 × 1 × 0 is 0.

0 × 1 × 0 = 0

Up 2: 1 × 2 × 1.

1 × 2 × 1 is 2.

1 × 2 × 1 = 2

Up 3: 4 × 3 × 2.

4 × 3 × 2 is 24.

4 × 3 × 2 = 24

4 Down minus up

D is all the down products, minus all the up products.

The brackets keep each group together.

D = (4 + 0 + 0) − (0 + 2 + 24)

Careful!

All three up products are taken away, not only the first.

That is why we add them up in brackets first.

Careful: not 4 − 0 + 2 + 24 ✗

The down products make 4 + 0 + 0 = 4.

The up products make 0 + 2 + 24 = 26.

D = 4 − 26

4 minus 26 is −22.

So D = −22.

D = −22

5 Check

Let's check the taking away: up plus D must give down.

26 + (−22) = 4.

So D = −22 is right!

26 + (−22) = 4

Let's look back.

First: copy the first two columns.

Then: multiply down.

Then: multiply up.

Then: down minus up.

Last: check.

Answer