1. Let $A = \begin{pmatrix}1 & 0 \\ -4 & 1 \\ \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & \frac{1}{b} \\ \end{pmatrix} \begin{pmatrix}1 & -2 \\ 0 & 1 \\ \end{pmatrix}$. Find the correct statement.

    $A^{-1} = \begin{pmatrix}1 & 0 \\ -4 & 1 \\ \end{pmatrix} \begin{pmatrix}1 & 0... ... \frac{1}{b} \\ \end{pmatrix} \begin{pmatrix}1 & -2 \\ 0 & 1 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & -2 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix}1 & 0... ... \frac{1}{b} \\ \end{pmatrix} \begin{pmatrix}1 & 0 \\ -4 & 1 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 0 \\ 4 & 1 \\ \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & b \\ \end{pmatrix} \begin{pmatrix}1 & 2 \\ 0 & 1 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 2 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & b \\ \end{pmatrix} \begin{pmatrix}1 & 0 \\ 4 & 1 \\ \end{pmatrix}$.

  2. Let $A$ and $B$ be $2\times 2$ matrices, and let $I$ be the $2\times 2$ identity matrix. Suppose $\begin{pmatrix}1 & 2 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix}1 & 0 \\ 2 & 1 ... ... \begin{pmatrix}\frac{1}{3} & 0 \\ 0 & 1 \\ \end{pmatrix} [A \, I] = [I \, B]$. Find $A$.

    $A = \begin{pmatrix}\frac{5}{3} & 2 \\ \frac{2}{3} & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}3 & 0 \\ -2 & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}15 & -2 \\ -6 & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}3 & -6 \\ -2 & 5 \\ \end{pmatrix}$.

  3. Let $A = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & 0 & 1 \\ \end{pmatrix}$. Find $A^{-1}$.

    $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ -a & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & a \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{1}{a} & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & 0 & 1 \\ \end{pmatrix}$

  4. Let $A = \begin{pmatrix}1 & 0 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \b... ...{pmatrix} \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{pmatrix}$. Find the correct statement.

    $A^{-1} = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{pmatrix} ... ...atrix} \begin{pmatrix}1 & 0 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ -4 & 0 & 1 \\ \end{pmatrix}... ...{pmatrix} \begin{pmatrix}1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} ... ...atrix} \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ -4 & 0 & 1 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 0 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}... ...{pmatrix} \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{pmatrix}$.

  5. Let $A^{-1} = \begin{pmatrix}1 & -3 & 2 \\ -1 & 4 & -4 \\ 0 & -2 & 5 \\ \end{pmatrix}$ and $b = \begin{pmatrix}-2 \\ -2 \\ 2 \\ \end{pmatrix}$ . Solve $A x = b$

    $\displaystyle \begin{pmatrix}0 \\ 2 \\ 2 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}8 \\ -14 \\ 14 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}8 \\ -6 \\ 2 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-38 \\ -16 \\ -6 \\ \end{pmatrix}$


Department of Mathematics
Last modified: 2025-12-16