Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 04/Solution 1

We reduce the matrix to row echelon form. Subtracting the first row from the last row transforms

$M=\left[\begin{array}{ccc}1& 7& 3\\ 0& p& 5\\ 1& pq+7& 8\end{array}\right]$

into

$\left[\begin{array}{ccc}1& 7& 3\\ 0& p& 5\\ 0& pq& 5\end{array}\right].$

Next, subtracting $q$ times the second row from the third row yields

$\left[\begin{array}{ccc}1& 7& 3\\ 0& p& 5\\ 0& 0& 5-5q\end{array}\right].$

Clearly, this matrix has rank at least 2 (since, for instance, the first and third columns are linearly independent regardless of the value of $q$). On the other hand, it must have rank less than 3 whenever $p=0$ or $5-5q=0$ (or both), which is to say that $M$ has rank 2 exactly when $p=0$ or $q=1$ (or both).