TITLE: Matrix Determinant question
QUESTION [0 upvotes]: $A$ is a $4 \times4$ matrix, $A$ has a typical element $a_{ij}$ s.t.
  $|A|=4$. Now consider a matrix $B$ with every $b_{ij}=-2\cdot a_{ij}$, then
  $|B|$ is ?

If we take out $-2$ from each row of $B$ we get $(-2)\cdot(-2)\cdot(-2)\cdot(-2)= 16$.
Therefore, $|B|=16\cdot 4=64$ ? Is this right?

REPLY [2 votes]: You are correct. In general:
If $A$ is an $n \times n$ matrix, then for $\lambda \in \mathbb{R}$:
$$|\lambda A| = \lambda^n|A|$$
In this case:
$$|B| =|-2A| = (-2)^4|A| = 16|A| = 64$$

REPLY [2 votes]: Yes, it is correct. In general if $A$ is a $n\times n$ matrix with coefficients in $K$ then $|\lambda A|=\lambda^n |A|$ for any $\lambda\in K$.
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