Find the determinant of a $3AA^t$

Question

I am trying to find the determinant of $3AA^t$ or $|3AA^t|$

where

$$A = \begin{bmatrix} -1 & 2 \\ 2 & 3\end{bmatrix}$$

So I wind up getting:

$$|A| = -3 - 4 = -7 = A^t$$

so $$|3AA^t| = -3 * -7 * -7 = 147$$

Is this right?

My check fails though. I try to find $3AA^t$ and I get this:

$$3AA^t = 3 * \begin{bmatrix} -1 & 2 \\ 2 & 3 \end{bmatrix} * \begin{bmatrix} -1 & 2 \\ 2 & 3 \end{bmatrix} $$

$$ =\begin{bmatrix} -3 & 6 \\ 6 & 9 \end{bmatrix} * \begin{bmatrix} -1 & 2 \\ 2 & 3 \end{bmatrix} $$

$$= \begin{bmatrix} -15 & 12 \\ 12 & 39 \end{bmatrix} $$

$$= 441$$

which does not $= 147$. Where did I go wrong?

Answer 1

If $A$ is an $n \times n$ matrix, $\det(3A) = \det(3I) \det(A) = 3^n \det(A)$, not $3 \det(A)$.

Answer 2

Because if $\lambda$ is a scalar and $A$ is a $n\times n$ matrix, then $\det(\lambda A)=\lambda^n\det A$. That's why you only got $\frac13$ of the right answer.

Answer 3

$3$ is a scalar and A is a $2×2$ matrix,then $det(3A)=3^2det(A)$.$$|3AA^t|=3^2×|AA^t|$$ $$|AA^t|=det \left( \begin{bmatrix} -1 & 2 \\ 2 & 3 \end{bmatrix} × \begin{bmatrix} -1 & 2 \\ 2 & 3 \end{bmatrix} \right)=det\left(\begin{bmatrix} 5 & 4 \\ 4 & 13\end{bmatrix} \right)=49$$ $|3AA^t|=9×49=441$.
Sorry if i do any mistake.