Question

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2. a. A is a 2 x 3 matrix whose columns span $R^2$. b. A is a 3 x 3 matrix whose columns span $R^3$. Example 1: The following matrices are in echelon form. The leading entries (?) may have any nonzero value; the starred entries () may have any value (including zero). $\begin{bmatrix} \Box & & * & * \\ 0 & \Box & * & * \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$, $\begin{bmatrix} 0 & \Box & * & * & * & * & * & * \\ 0 & 0 & 0 & \Box & * & * & * & * \\ 0 & 0 & 0 & 0 & 0 & \Box & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Box \end{bmatrix}$ The following matrices are in reduced echelon form because the leading entries are 1's, and there are 0's below and above each leading 1. $\begin{bmatrix} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$, $\begin{bmatrix} 0 & 1 & * & * & * & * & 0 & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$

          Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section
1.2.
a. A is a 2 x 3 matrix whose columns span $R^2$.
b. A is a 3 x 3 matrix whose columns span $R^3$.
Example 1: The following matrices are in echelon form. The leading entries (?) may have any
nonzero value; the starred entries (*) may have any value (including zero).
$\begin{bmatrix}
\Box & * & * & * \\
0 & \Box & * & * \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}$, $\begin{bmatrix}
0 & \Box & * & * & * & * & * & * \\
0 & 0 & 0 & \Box & * & * & * & * \\
0 & 0 & 0 & 0 & 0 & \Box & * & * \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \Box
\end{bmatrix}$
The following matrices are in reduced echelon form because the leading entries are 1's, and
there are 0's below and above each leading 1.
$\begin{bmatrix}
1 & 0 & * & * \\
0 & 1 & * & * \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}$, $\begin{bmatrix}
0 & 1 & * & * & * & * & 0 & * \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & * \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}$

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section
1.2.
a. A is a 2 x 3 matrix whose columns span R^2.
b. A is a 3 x 3 matrix whose columns span R^3.
Example 1: The following matrices are in echelon form. The leading entries (?) may have any
nonzero value; the starred entries (*) may have any value (including zero).
< b m a t r i x >, < b m a t r i x >
The following matrices are in reduced echelon form because the leading entries are 1's, and
there are 0's below and above each leading 1.
< b m a t r i x >, < b m a t r i x >

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