If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?
Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.
Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.
Otherwise, provide a counterexample.
Proof. […]

Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
(b) Find all such matrices with rank 2.
Solution.
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
First we look at the rank 1 case. […]

If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix
Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.
Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.
Proof.
Because $A$ has rank $n$, we know that the $n \times n$ […]

If Two Matrices Have the Same Rank, Are They Row-Equivalent?
If $A, B$ have the same rank, can we conclude that they are row-equivalent?
If so, then prove it. If not, then provide a counterexample.
Solution.
Having the same rank does not mean they are row-equivalent.
For a simple counterexample, consider $A = […]

Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank
For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.
(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.
(c) $C […]

For Which Choices of $x$ is the Given Matrix Invertible?
Determine the values of $x$ so that the matrix
\[A=\begin{bmatrix}
1 & 1 & x \\
1 &x &x \\
x & x & x
\end{bmatrix}\]
is invertible.
For those values of $x$, find the inverse matrix $A^{-1}$.
Solution.
We use the fact that a matrix is invertible […]

Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
(a) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
[…]