Det A = 6, find det(A^T)

Step by step, with the reason for every step · Determinant rules · Linear Algebra

Step by step

Question: find det(A transpose), when det A = 6.

det A = 6

A transpose is A with its rows written as columns.

It has the same determinant as A.

Aᵀ: rows become columns

det(Aᵀ) = det A

So det of A transpose is det A.

det(A^T) = det A

Careful!

Turning rows into columns does not change the determinant, not even its sign.

Careful: rows ↔ columns: D does not change

So det(A transpose) = 6.

det(A^T) = 6

Let's check with an easy A: its determinant is 6 × 1 − 0 × 1 = 6.

A = [6, 1; 0, 1]

Rows become columns.

A^T = [6, 0; 1, 1]

Down minus up: 6.

det(A^T) = 6 × 1 − 1 × 0 = 6

We get 6 again.

So the answer is right!

Let's look back.

First: use the rule.

Then: work it out.

Last: check.

Answer

det(A^T) = 6

Try this one yourself, then check:

det A = 2, find det(A^T)

Show the answer

det(A^T) = 2