Science:Math Exam Resources/Courses/MATH221/April 2010/Question 05/Solution 1

The answer is false.

Let $A$ be the 2-by-2 identity matrix. Then

$\det(-A)=\det(-I)=\det {\begin{bmatrix}-1&0\\0&-1\end{bmatrix}}=(-1)(-1)-(0)(0)=1$

whereas

$-\det(A)=-\det(I)=-\det {\begin{bmatrix}1&0\\0&1\end{bmatrix}}=-[(1)(1)-(0)(0)]=-1.$