## Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?

## Problem 669

**(a)** Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular?

**(b)** Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system nonsingular?

**(c)** Let $A$ be a $4\times 4$ matrix and let

\[\mathbf{v}=\begin{bmatrix}

1 \\

2 \\

3 \\

4

\end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix}

4 \\

3 \\

2 \\

1

\end{bmatrix}.\]
Suppose that we have $A\mathbf{v}=A\mathbf{w}$. Is the matrix $A$ nonsingular?