## 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}$.

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.

Express the vector $\mathbf{b}=\begin{bmatrix}

2 \\

13 \\

6

\end{bmatrix}$ as a linear combination of the vectors

\[\mathbf{v}_1=\begin{bmatrix}

1 \\

5 \\

-1

\end{bmatrix},

\mathbf{v}_2=

\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix},

\mathbf{v}_3=

\begin{bmatrix}

1 \\

4 \\

3

\end{bmatrix}.\]

(*The Ohio State University, Linear Algebra Exam*)

Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.

**(a)** \[\left\{

\begin{array}{c}

ax+by=c \\

dx+ey=f,

\end{array}

\right.

\]
where $a,b,c, d$ are scalars satisfying $a/d=b/e=c/f$.

**(b)** $A \mathbf{x}=\mathbf{0}$, where $A$ is a singular matrix.

**(c)** A homogeneous system of $3$ equations in $4$ unknowns.

**(d) **$A\mathbf{x}=\mathbf{b}$, where the row-reduced echelon form of the augmented matrix $[A|\mathbf{b}]$ looks as follows:

\[\begin{bmatrix}

1 & 0 & -1 & 0 \\

0 &1 & 2 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}.\]
(*The Ohio State University, Linear Algebra Exam*)

Read solution

Solve the following system of linear equations using Gaussian elimination.

\begin{align*}

x+2y+3z &=4 \\

5x+6y+7z &=8\\

9x+10y+11z &=12

\end{align*}