Question

If A and B ae both non-singular $n\times n$ matrices, then which of the following is Not True?

• A. $det(A.B)=\left(detA\right).\left(detB\right)$
• B. $det\left(A^{-1}\right)=(detA{)}^{-1}$
• C. $det\left(A^T\right)=detA$
• D. $det(A+B)=det\left(A\right)+det\left(B\right)$

Answer 1

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

Let takes a example:
A=$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$and B=$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

det (A)=1 det (B)=4

$AB=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

Now, det(A.B) = 4 = det(A). det(B)

So statement (a) is true

$A^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]det.\left(A^{-1}\right)=1$

$(detA{)}^{-1}=1=det(A^{-1})$

So statement (b) is true

$A^T=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],det\left(A^T\right)=1=detA$

So statement (c) is true

$A+B=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right],det(A+B)=9$

$det(A+B)=9\ne det\left(A\right)+det\left(B\right)$

So statement (d) is not true.