In Exercises C30–C33 determine if the matrix is nonsingular or singular. Give reasons for your answer.
\begin{equation*} \begin{bmatrix} -3 & 1 & 2 & 8\\ 2 & 0 & 3 & 4\\ 1 & 2 & 7 & -4\\ 5 & -1 & 2 & 0 \end{bmatrix} \end{equation*}
\begin{equation*} \begin{bmatrix} 2 & 3 & 1 & 4\\ 1 & 1 & 1 & 0\\ -1 & 2 & 3 & 5\\ 1 & 2 & 1 & 3 \end{bmatrix} \end{equation*}
\begin{equation*} \begin{bmatrix} 9 & 3 & 2 & 4\\ 5 & -6 & 1 & 3\\ 4 & 1 & 3 & -5 \end{bmatrix} \end{equation*}
\begin{equation*} \begin{bmatrix} -1 & 2 & 0 & 3 \\ 1 & -3 & -2 & 4 \\ -2 & 0 & 4 & 3 \\ -3 & 1 & -2 & 3 \end{bmatrix} \end{equation*}
Each of the archetypes below is a system of equations with a square coefficient matrix, or is itself a square matrix. Determine if these matrices are nonsingular, or singular. Comment on the null space of each matrix.
Archetype A,
Archetype B,
Archetype F,
Archetype K,
Archetype L
Find the null space of the matrix $E$ below. \begin{align*} E&= \begin{bmatrix} 2 & 1 & -1 & -9 \\ 2 & 2 & -6 & -6 \\ 1 & 2 & -8 & 0 \\ -1 & 2 & -12 & 12 \end{bmatrix} \end{align*}
Let $A$ be the coefficient matrix of the system of equations below. Is $A$ nonsingular or singular? Explain what you could infer about the solution set for the system based only on what you have learned about $A$ being singular or nonsingular. \begin{align*} -x_1+5x_2&=-8\\ -2x_1+5x_2+5x_3+2x_4&=9\\ -3x_1-x_2+3x_3+x_4&=3\\ 7x_1+6x_2+5x_3+x_4&=30 \end{align*}
For Exercises M51–M52 say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions.
A system with a nonsingular coefficient matrix, not homogeneous.
Suppose that $A$ is a square matrix, and $B$ is a matrix in reduced row-echelon form that is row-equivalent to $A$. Prove that if $A$ is singular, then the last row of $B$ is a zero row.
Using (Definition RREF) and (Definition IM) carefully, give a proof of the following equivalence: $A$ is a square matrix in reduced row-echelon form where every column is a pivot column if and only if $A$ is the identity matrix.
Suppose that $A$ is a nonsingular matrix and $A$ is row-equivalent to the matrix $B$. Prove that $B$ is nonsingular.
Suppose that $A$ is a square matrix of size $n\times n$ and that we know there is a single vector $\vect{b}\in\complex{n}$ such that the system $\linearsystem{A}{\vect{b}}$ has a unique solution. Prove that $A$ is a nonsingular matrix. (Notice that this is very similar to Theorem NMUS, but is not exactly the same.)
Provide an alternative for the second half of the proof of Theorem NMUS, without appealing to properties of the reduced row-echelon form of the coefficient matrix. In other words, prove that if $A$ is nonsingular, then $\linearsystem{A}{\vect{b}}$ has a unique solution for every choice of the constant vector $\vect{b}$. Construct this proof without using Theorem REMEF or Theorem RREFU.