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

    $A^{-1} = \begin{pmatrix}-2 & \frac{3}{2} \\ -1 & \frac{1}{2} \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}-11 & -6 \\ 4 & 2 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 3 \\ -2 & -\frac{11}{2} \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & -3 \\ 2 & -4 \\ \end{pmatrix}$.

  2. Let $A = \begin{pmatrix}-2 & 3 \\ 4 & -8 \\ \end{pmatrix}$. Find $A^{-1}$ if possible.

    $\displaystyle$   Does not exist $\displaystyle \begin{pmatrix}2 & \frac{3}{4} \\ 1 & \frac{1}{2} \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-2 & -\frac{3}{4} \\ -1 & -\frac{1}{2} \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-8 & -3 \\ -4 & -2 \\ \end{pmatrix}$

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

    $A = \begin{pmatrix}-\frac{7}{2} & -3 \\ 1 & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}-2 & -6 \\ 2 & 7 \\ \end{pmatrix}$. $A = \begin{pmatrix}-14 & 3 \\ -4 & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}-2 & 0 \\ 2 & 1 \\ \end{pmatrix}$.

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

    $\displaystyle \begin{pmatrix}3 \\ 17 \\ -3 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-3 \\ -4 \\ -3 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}-3 \\ -4 \\ -3 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}3 \\ 8 \\ -3 \\ \end{pmatrix}$

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

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


Department of Mathematics
Last modified: 2025-09-25