If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent?

Problem 695

Consider the following system of linear equations:
\begin{align*}
ax_1+bx_2 &=c\\
dx_1+ex_2 &=f\\
gx_1+hx_2 &=i.
\end{align*}

(a) Write down the augmented matrix.

(b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? Justify your answer.

Solution.

(a) Write down the augmented matrix.

The augmented matrix of the system is
$\left[\begin{array}{rr|r} a & b & c \\ d &e &f \\ g & h & i \end{array}\right].$

(b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? Justify your answer.

Since the augmented matrix is row equivalent, we have
$\left[\begin{array}{rr|r} a & b & c \\ d &e &f \\ g & h & i \end{array}\right] \xrightarrow{\text{elementary row operations}} \left[\begin{array}{rr|r} 1 & 0 & 0 \\ 0 &1 &0 \\ 0 & 0 & 1 \end{array}\right].$ Thus the third row corresponds to the equation $0x_1+0x_2=1$, equivalently, $0=1$.
This yields that the system is inconsistent.

Common Mistake

This is a midterm exam problem of Lienar Algebra at the Ohio State University.

For part (b), many students wrote that “the identity matrix is nonsingular, so it is consistent”.
Well, if the coefficient matrix of a system is row equivalent to the identity, then this is ture but in our case, the augmented matrix is row-equivalent to the identity matrix.

Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression $C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,$ which matrix do you get?...