How to prove the value of a "negative" matrix

Question

So I came across the formula that $\det(-A)=(-1)^n \cdot det(A)$, where $n$ is the number of columns/rows of A. I know how you get the formula by Laplace's formula and only described in words somehow, but I would also like a formal proof of it.

Its better to think of it as $\det(cA)=c^n\cdot\det(A)$ – Jan 10 '16 at 23:53 — , Jan 10 '16 at 23:53

score 2 · Accepted Answer · answered Jan 10 '16 at 23:57

2

$-A=(-I)A=\begin{pmatrix}-1&0&\dots&0\\0&-1&\dots&0\\\vdots&\vdots&\ddots&\vdots\\0&0&\dots&-1\end{pmatrix}A,\;$ so $\det(-A)=\det(-I)\det A=(-1)^n\det A$.

answered Jan 10 '16 at 23:57

Bernard

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How to prove the value of a "negative" matrix

1 Answers1