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A 3 × 3 matrix A has |A| = -2. Find |3A|, |A²|, |-A| and |A^TA|.

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A $3 \times 3$ matrix $A$ has $|A| = -2$ . Find $|3A|$ , $|A^2|$ , $|-A|$ and $|A^TA|$ .

Need a reminder? The key ideaShow

Swapping two rows changes the sign. Multiplying a row by $k$ multiplies the determinant by $k$ . Adding a multiple of one row to another changes nothing. Equal or proportional rows give $0$ . Also $|kA| = k^n|A|$ and $|AB| = |A||B|$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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