## If Two Matrices Have the Same Rank, Are They Row-Equivalent?

## Problem 644

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.

Add to solve laterIf $A, B$ have the same rank, can we conclude that they are row-equivalent?

If so, then prove it. If not, then provide a counterexample.

Add to solve laterFor 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 \\

0 & 1 & -1

\end{bmatrix}$

**(b)** $A=\begin{bmatrix}

1 & 0 & 2 \\

-1 &-3 &2 \\

3 & 6 & -2

\end{bmatrix}$.

Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination).

Find the vector form for the general solution.

\begin{align*}

x_1-x_3-3x_5&=1\\

3x_1+x_2-x_3+x_4-9x_5&=3\\

x_1-x_3+x_4-2x_5&=1.

\end{align*}

Find the rank of the following real matrix.

\[ \begin{bmatrix}

a & 1 & 2 \\

1 &1 &1 \\

-1 & 1 & 1-a

\end{bmatrix},\]
where $a$ is a real number.

(*Kyoto University, Linear Algebra Exam*)

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