Exercise 5

Determine whether the following statements are true or false and explain your response.

1. If we have a square matrix $A$ and multiply the first row by $5$ and add it to the third row to obtain $A'$, then $\det(A') = 5\det(A)$.

2. If we interchange two rows of a matrix, then the determinant is unchanged.

3. If we scale a row of the matrix $A$ by $17$ to obtain $A'$, then $\det(A') = 17\det(A)$.

4. If $A$ and $A'$ are row equivalent and $\det(A') = 0$, then $\det(A) = 0$ also.

5. If $A$ is row equivalent to the identity matrix, then $\det(A) = \det(I) = 1$.