Question

If A and B are square matrices of size n$\times$n, then which of the following statements is not true.

• A. $det\left(AB\right)=det\left(A\right)det\left(B\right)$
• B. $det\left(kA\right)=k^{n}det\left(A\right)$
• C. $det(A+B)=det\left(A\right)+det\left(B\right)$
• D. $det\left(A^T\right)=\frac{1}{det\left(A^{-1}\right)}$

Answer 1

The correct option is C $det(A+B)=det\left(A\right)+det\left(B\right)$

(a) $\left|AB\right|=\left|A\right|\left|B\right|$
it is true
(b) $\left|kA\right|$=$k^n$$\left|A\right|$
Where n is order of matrix
(c) $|A+B|=\left|A\right|+\left|B\right|$
it is false
(d) $\left|A^T\right|=\left|A\right|=\frac{1}{\left|A^{-1}\right|}$,which is true.
Hence (C) is the correct answer .