## Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2

## Problem 646

**(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.

**(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.

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 consists of columns of $A$. For each columns, $A_j$ of $A$ that does not appear in the basis, express $A_j$ as a linear combination of the basis vectors.

**(d)** Exhibit a basis for the row space of $A$.

Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.

\[A= \left[\begin{array}{rrr|r}

1 & 2 & 3 & 4 \\

2 &-1 & -2 & a^2 \\

-1 & -7 & -11 & a

\end{array} \right],\]
where $a$ is a real number. Determine all the values of $a$ so that the corresponding system is consistent.

Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where

\begin{align*}

A_1=\begin{bmatrix}

1 & 2 \\

-1 & 3

\end{bmatrix}, \quad

A_2=\begin{bmatrix}

0 & -1 \\

1 & 4

\end{bmatrix}, \quad

A_3=\begin{bmatrix}

-1 & 0 \\

1 & -10

\end{bmatrix}, \quad

A_4=\begin{bmatrix}

3 & 7 \\

-2 & 6

\end{bmatrix}.

\end{align*}

Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.