Step by step
The question
Question: find det(3A), when det A = 2 and A is 2 by 2.
det A = 2, A is 2 × 2
1 Use the rule
3A means every entry times 3.
So all 2 rows are times 3.
Each row times 3 makes the determinant times 3.
3A: every entry × 3, so every row × 3
each row × 3: D × 3
One 3 for each row: 2 of them.
det(3A) = 3 × 3 × det A
Careful!
Not just one 3: every row brings its own 3.
That is 2 of them.
Careful: not 3 × 2 = 6 ✗
2 Work it out
3 × 3 is 3 to the power 2.
det(3A) = 32 × 2
3 to the power 2 is 9.
det(3A) = 9 × 2
So det(3A) = 18.
det(3A) = 18
3 Check
Let's check with an easy A that has det A = 2: 2 and 1s down the diagonal, 0s elsewhere.
A = [NaN, 0; 0, 1]
Its determinant is the diagonal multiplied: 2.
Good.
det A = NaN × 1 = NaN
3A: every entry times 3.
3A = [NaN, 0; 0, 3]
Its determinant: the diagonal multiplied, 18.
det (3A) = NaN × 3 = NaN
We get 18 again.
So the answer is right!
Let's look back.
First: use the rule.
Then: work it out.
Last: check.
Answer
det(3A) = 18