Assignment 19: More matrix equations and inverses

1. Show that (\begin{matrix}1 & 2 & 3 & 4 & 5 \\ 6 & 7 & 8 & 9 & 1 \\ 2 & 3 & 4 & 5 & 6 \\ 7 & 8 & 9 & 1 & 2 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}) = (\begin{matrix}b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}) is solvable for all choice of b_{i}. Hint: the

row-reduced echelon form of the 4 × 5 matrix is R = (\begin{matrix}1 & 0 & −1 & 0 & 0 \\ 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}). Moreover, show that for each choice of (\begin{matrix}b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}), there are infinitely many solutions.

2. Find all (\begin{matrix}b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \\ b_{5} \end{matrix}) for which

(\begin{matrix}1 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 5 & 0 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}) = (\begin{matrix}b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \\ b_{5} \end{matrix}) is solvable.

Find all solutions to (\begin{matrix}1 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 5 & 0 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}) = (\begin{matrix}1 \\ 2 \\ 3 \\ 4 \\ 5 \end{matrix})

3. Let A = (\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 4 \end{matrix}).

Use row reduction to find the inverse of A, and solve the matrix equation Ax = (\begin{matrix}1 \\ 2 \\ 3 \end{matrix}) both by using the inverse of A, and by row reducing the augmented matrix.