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.

Possibilities For the Number of Solutions for a Linear System
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\]
where $a,b,c, d$ […]

Summary: Possibilities for the Solution Set of a System of Linear Equations
In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.
Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ […]

Solving a System of Linear Equations By Using an Inverse Matrix
Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}
(a) Find the coefficient matrix and its inverse matrix.
(b) Using the inverse matrix, solve the system of linear equations.
(The Ohio […]

Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$
Let $A$ be the coefficient matrix of the system of linear equations
\begin{align*}
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
\end{align*}
(a) Solve the system by finding the inverse matrix $A^{-1}$.
(b) Let $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ be the solution […]

Linear Algebra Midterm 1 at the Ohio State University (1/3)
The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.
There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).
The time limit was 55 minutes.
This post is Part 1 and contains the […]

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?...