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

    $\displaystyle \begin{pmatrix}1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ -4 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$ $\displaystyle \begin{pmatrix}1 & 0 & -\frac{1}{4} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$

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

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

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

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

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

    $A^{-1} = \begin{pmatrix}1 & -3 \\ 3 & -\frac{17}{2} \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}1 & 3 \\ -3 & -7 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}-17 & 6 \\ -6 & 2 \\ \end{pmatrix}$. $A^{-1} = \begin{pmatrix}-\frac{7}{2} & -\frac{3}{2} \\ \frac{3}{2} & \frac{1 }{2} \\ \end{pmatrix}$.

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

    $A = \begin{pmatrix}-3 & 0 \\ -3 & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}-21 & -2 \\ 9 & 1 \\ \end{pmatrix}$. $A = \begin{pmatrix}-3 & 6 \\ -3 & 7 \\ \end{pmatrix}$. $A = \begin{pmatrix}-\frac{7}{3} & 2 \\ -1 & 1 \\ \end{pmatrix}$.


Department of Mathematics
Last modified: 2025-07-21