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

$M={\begin{bmatrix}1&7&3\\0&p&5\\1&pq+7&8\end{bmatrix}}$

into

${\begin{bmatrix}1&7&3\\0&p&5\\0&pq&5\end{bmatrix}}.$

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

${\begin{bmatrix}1&7&3\\0&p&5\\0&0&5-5q\end{bmatrix}}.$

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 $\color {blue}p=0$ or $\color {blue}q=1$ (or both).