Exercise 7.5.13

Specify whether each statement is true or false. If true, provide a proof. If false, provide a counterexample.

1. If $A$ is a $3\times 3$-matrix with determinant zero, then one column must be a multiple of some other column.

2. If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.

3. For two $n\times n$-matrices $A$ and $B$, $\det(A+B)=\det(A)+\det(B)$.

4. For an $n\times n$-matrix $A$, $\det(3A)=3\det(A)$.

5. If $A^{-1}$ exists, then $\det(A^{-1})=\det(A)^{-1}$.

6. If $B$ is obtained by multiplying a single row of $A$ by $4$, then $\det(B)=4\det(A)$.

7. For an $n\times n$-matrix $A$, we have $\det(-A)=(-1)^{n}\det(A)$.

8. If $A$ is a real $n\times n$-matrix, then $\det(A^{T}A)\geq 0$.

9. If $A^{k}=0$ for some positive integer $k$, then $\det(A)=0$.

10. If $A\mathbf{x}=0$ for some $\mathbf{x}\neq 0$, then $\det(A)=0$.