0.1 Practice Problems

1.

Show that for an invertible square matrix \(A\), the system \(Ax = b\) is always consistent.

2.

Show that the following system cannot have a unique solution for all \(k \in \mathbb {R}\). \[ \begin {bmatrix} 1 & 2 \\ 2 & 4 \end {bmatrix} \begin {bmatrix} x \\ y \end {bmatrix} = \begin {bmatrix} 1 \\ k \end {bmatrix} \]

3.

Use Gaussian Elimination to find the solution for the following system. \begin{align*} x + y + z &= 10 \\ 3x + 4y - z &= 2 \\ -x - 3y + z &= 4 \end{align*}

4.

Solve the same system above using LU decomposition.

5.

Find the inverse of the following matrix. \[ \begin {bmatrix} 1 & 1 & 1 \\ 3 & 4 & -1 \\ -1 & -3 & 1 \end {bmatrix} \] Then, find the values of \(x, y, z\) that satisfy the following system. \[ \begin {bmatrix} 1 & 1 & 1 \\ 3 & 4 & -1 \\ -1 & -3 & 1 \end {bmatrix} \begin {bmatrix} x \\ y \\ z \end {bmatrix} = \begin {bmatrix} 10 \\ 2 \\ 4 \end {bmatrix} \]