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 later Tagged: reduced echelon form
Add to solve later Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
Problem 552
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) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
0 & 1 & -1
\end{bmatrix}$
(b) $A=\begin{bmatrix}
1 & 0 & 2 \\
-1 &-3 &2 \\
3 & 6 & -2
\end{bmatrix}$.
Add to solve later Vector Form for the General Solution of a System of Linear Equations
Problem 267
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*}
Add to solve later Find the Rank of a Matrix with a Parameter
Problem 103
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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