1. Let $A = \begin{pmatrix}0 & a_{1,2} & a_{1,3} & a_{1,4} \\ 0 & a_{2,2} & a_{ 2,3} &... ...{3,2} & a_{3,3} & a_{3,4} \\ 0 & a_{4,2} & a_{4,3} & a_{4,4} \\ \end{pmatrix}$. Suppose that $\det A_{ 3,1} = -4$. Then find $\det A$.

    12 −12 −3 −4 3

  2. Express $\displaystyle x_{1}$ of the solution to the matrix equation $\displaystyle \begin{pmatrix}3 & 4 & s \\ 4 & -3 & -2 \\ -2 & s & 3 \\ \end{p... ...x_{2} \\ x_{3} \\ \end{pmatrix} = \begin{pmatrix}3 \\ -2 \\ -2 \\ \end{pmatrix}$.

    $\displaystyle -\frac{2}{4\,s^2-59}$ $\displaystyle -\frac{12\,s+54}{4\,s^2-59}$ $\displaystyle \frac{18\,s+48}{4\,s^2-59}$ $\displaystyle -\frac{2\,s^2-13}{4\,s^2-59}$ $\displaystyle \frac{3}{4\,s^2-59}$

  3. Let $E = \begin{pmatrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix}$. Find $\det E$.

    1 0 −1

  4. Let $A = \begin{pmatrix}3 & s & -2 \\ -2 & 4 & s \\ 3 & 2 & 3 \\ \end{pmatrix}$. Find $\det A$.

    $\displaystyle 3\,s^2+68$ −15 $\displaystyle -3\,s^2-68$ 15 2s − 12 12 − 2s

  5. Let $A = \begin{pmatrix}0 & k & v & w \\ 0 & 0 & -3 & z \\ 0 & 0 & 0 & -1 \\ -1 & s & t & u \\ \end{pmatrix}$. Find $\det A$.

    3k −3ku −3k −3k 3ku 3k


Department of Mathematics
Last modified: 2026-11-01