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 \(A'\text{,}\) then \(\det A' = 5\det A\text{.}\)

  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 \(A'\text{,}\) then \(\det A' = 17\det A\text{.}\)

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

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