Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 02 (d)/Solution 1

By (b), a row echelon form of the augmented matrix is

$\left({\begin{array}{ccc|c}1&1&1&5\\0&-1&0&-4\\0&0&1-a&b-a\end{array}}\right).$

The system has infinitely many solutions if the row echelon form has more variables than non-zero rows.

We have 3 variables $x,y,z$. Note that first two rows cannot be zero rows because of the nonzero pivots.

In order for the system to have infinitely many solutions, the third row must be a zero row.

Therefore, we need $1-a=0$ and $b-a=0$, so that the answer is $\color {blue}a=b=1$.