Det A = 5, A is 3 × 3, find det(2A)

Step by step

The question

Question: find det(2A), when det A = 5 and A is 3 by 3.

1 Use the rule

2A means every entry times 2.

So all 3 rows are times 2.

Each row times 2 makes the determinant times 2.

One 2 for each row: 3 of them.

Careful!

Not just one 2: every row brings its own 2.

That is 3 of them.

2 Work it out

2 × 2 × 2 is 2 to the power 3.

2 to the power 3 is 8.

So det(2A) = 40.

3 Check

Let's check with an easy A that has det A = 5: 5 and 1s down the diagonal, 0s elsewhere.

Its determinant is the diagonal multiplied: 5.

Good.

2A: every entry times 2.

Its determinant: the diagonal multiplied, 40.

We get 40 again.

So the answer is right!

Let's look back.

First: use the rule.

Then: work it out.

Last: check.