Consider the following system of linear equations:
\begin{align*}
ax_1+bx_2 &=c\\
dx_1+ex_2 &=f\\
gx_1+hx_2 &=i.
\end{align*}
(a) Write down the augmented matrix.
(b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? Justify your answer.
Add to solve later 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 For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.
(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.
(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.
(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.
(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.
Add to solve later If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?
If so, then prove it. If not, then provide a counterexample.
Add to solve later 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 $A$.
Add to solve later We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.
Let $A$ and $I$ be $2\times 2$ matrices defined as follows.
\[A=\begin{bmatrix}
1 & b\\
c& d
\end{bmatrix}, \qquad I=\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix}.\]
Prove that the matrix $A$ is row equivalent to the matrix $I$ if $d-cb \neq 0$.
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