If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?
If so, then prove it. If not, then provide a counterexample.
Two matrices are said to be row equivalent if one can be obtained from the other by a sequence of elementary row operations.
Proof.
Yes, in this case $A$ and $C$ are row-equivalent.
By assumption, the matrices $A$ and $B$ are row-equivalent, which means that there is a sequence of elementary row operations that turns $A$ into $B$.
Call this sequence $r_1 , r_2 , \cdots , r_n$, where each $r_i$ is an elementary row operation.
(Start with applying $r_1$ to $A$.)
By another assumption, $B$ is row-equivalent to $C$, which means that there is a sequence of elementary row operations which transforms $B$ into $C$; call this sequence $s_1 , s_2 , \cdots , s_m$.
Putting these sequences together, the operations $r_1 , r_2 , \cdots , r_n$ , $s_1 , s_2 , \cdots , s_m$ will transform the matrix $A$ into $C$.
Condition that Two Matrices are Row Equivalent
We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.
Let $A$ and $I$ be $2\times 2$ matrices defined as follows.
\[A=\begin{bmatrix}
1 & b\\
c& d
\end{bmatrix}, \qquad […]
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 […]
Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix
Let \[A=\begin{bmatrix}
1 & 1 & 2 \\
2 &2 &4 \\
2 & 3 & 5
\end{bmatrix}.\]
(a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$.
(b) Find a basis for the null space of $A$.
(c) Find a basis for the range of $A$ that […]
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 = […]
If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent?
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? […]
The Inverse Matrix of an Upper Triangular Matrix with Variables
Let $A$ be the following $3\times 3$ upper triangular matrix.
\[A=\begin{bmatrix}
1 & x & y \\
0 &1 &z \\
0 & 0 & 1
\end{bmatrix},\]
where $x, y, z$ are some real numbers.
Determine whether the matrix $A$ is invertible or not. If it is invertible, then find […]
Find Values of $a$ so that the Matrix is Nonsingular
Let $A$ be the following $3 \times 3$ matrix.
\[A=\begin{bmatrix}
1 & 1 & -1 \\
0 &1 &2 \\
1 & 1 & a
\end{bmatrix}.\]
Determine the values of $a$ so that the matrix $A$ is nonsingular.
Solution.
We use the fact that a matrix is nonsingular if and only if […]