Step by step
The question
Question: find the determinant of this matrix.
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.
2 Multiply down
Three diagonals go down to the right.
We multiply the three numbers on each one.
Down 1: 1 × 4 × 5.
1 × 4 × 5 is 20.
Down 2: 2 × 1 × 2.
2 × 1 × 2 is 4.
Down 3: 3 × 0 × 1.
3 × 0 × 1 is 0.
3 Multiply up
Three diagonals go up to the right.
We multiply those too.
Up 1: 2 × 4 × 3.
2 × 4 × 3 is 24.
Up 2: 1 × 1 × 1.
1 × 1 × 1 is 1.
Up 3: 5 × 0 × 2.
5 × 0 × 2 is 0.
4 Down minus up
D is all the down products, minus all the up products.
The brackets keep each group together.
Careful!
All three up products are taken away, not only the first.
That is why we add them up in brackets first.
The down products make 20 + 4 + 0 = 24.
The up products make 24 + 1 + 0 = 25.
24 minus 25 is −1.
So D = −1.
5 Check
Let's check the taking away: up plus D must give down.
25 + (−1) = 24.
So D = −1 is right!
Let's look back.
First: copy the first two columns.
Then: multiply down.
Then: multiply up.
Then: down minus up.
Last: check.