Assignment 19: More matrix equations and inverses

1.

Show that $\begin{pmatrix}1&2&3&4&5\\ 6&7&8&9&1\\ 2&3&4&5&6\\ 7&8&9&1&2\end{pmatrix}\begi... ...\ x_4\\ x_5 \end{pmatrix} = \begin{pmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{pmatrix}$ is solvable for all choice of $b_i$. Hint: the row-reduced echelon form of the $4\times 5$ matrix is $R = \begin{pmatrix}1&0&-1&0&0\\ 0&1&2&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{pmatrix}$.

Moreover, show that for each choice of $\begin{pmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{pmatrix}$, there are infinitely many solutions.

2.

Find all $\begin{pmatrix}b_1\\ b_2\\ b_3\\ b_4\\ b_5 \end{pmatrix}$ for which

$\begin{pmatrix}1&0&0&0\\ 2&0&0&0\\ 0&3&0&0\\ 0&4&0&0\\ 0&0&5&0\end{pmatrix}\be... ...\ x_4 \end{pmatrix} = \begin{pmatrix} b_1\\ b_2\\ b_3\\ b_4\\ b_5 \end{pmatrix}$ is solvable. Find all solutions to $\begin{pmatrix}1&0&0&0\\ 2&0&0&0\\ 0&3&0&0\\ 0&4&0&0\\ 0&0&5&0\end{pmatrix}\be... ... x_2\\ x_3\\ x_4 \end{pmatrix} =\begin{pmatrix} 1\\ 2\\ 3\\ 4\\ 5 \end{pmatrix}$

3.

Let $A = \begin{pmatrix}1&0&1\\ 0&1&2\\ 1&2&4\end{pmatrix}$. Use row reduction to find the inverse of $A$, and solve the matrix equation $A {\mathbf x}= \begin{pmatrix}1\\ 2\\ 3 \end{pmatrix}$ both by using the inverse of $A$, and by row reducing the augmented matrix.