Which of the following statements is always TRUE
A. If $A$ is an $n\times n$ matrix with all entries being positive, then $\det(A)>0$.
B. If $A$ and $B$ are two $n\times n$ matrices with $\det(A)>0$ and $\det(B)>0$, then also $\det(A+B)>0$.
C. If $A$ and $B$ are two $n\times n$ matrices such that $AB=0$, then both $A$ and $B$ are singular.
D. If rows of an $n\times n$ matrix $A$ are linearly independent, then $\det(A^{T}A)>0$.
E. If $A$ is an $n\times n$ matrix with $A^2=I_n$, then $\det(A)=1$.
My analysis:
A - this is obvious false. Easy to find a $2\times 2$ matrix example.
B - I am not sure.
C - this is false as $B$ could be zero matrix.
D - Use row reduction, we can get $\det(A)=0$. And $\det(A^{T}A)=\det(A^{T})\det(A)=(\det(A))^2=0$. So it is false.
E - $\det(A^2)=(\det(A))^2=(\det(I_n))^2=1$, so $\det(A)=1$ or $\det(A)=-1$. It is false.
Please check if my analysis is correct.
Source of this question - it is the problem 5 of Purdue MA26500 (Linear Algebra) Fall 2022 Midterm 1.
A) $\begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}$
B) $\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & -1\end{pmatrix}$
C) A non singular, B zero matrix.
D) True. $\det(A^T A) = \det(A^T) \det A = (\det A)^2 > 0$, because $\det A$ is nonzero.
E) $\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$