Chapter 1.9, Problem 78E

Use Theorem $3$ to prove Theorem $16$.

THEOREM 3: Let $\left[C\right. \left|\left.\mathit{ }\mathit{d}\right]\right.$ be an $\left[m\times (n+1)\right]$ matrix in reduced echelon form, where $\left[C\right. \left|\left.\mathit{ }\mathit{d}\right]\right.$ represents a consistent system. Let $\left[C\right. \left|\left.\mathit{ }\mathit{d}\right]\right.$ have $r$ nonzero rows. Then $r\le n$ and in the solution of the system there are $n-r$ variables that can be assigned arbitrary values.

THEOREM 16: Let $A$ be an $(n\times n)$ marix. Then $A$ is nonsingular if and only if $A$ is row equivalent to $I$.