Question 1

Use the properties of determinants to find the value of the second determinant, given the value of the first.

Use the properties of determinants to find the value of the second determinant, given-example-1

Question 2

$det\begin{bmatrix}{s} & {t} & {u} \\ {v} & {w} & {x} \\ {4} & {2} & {8}\end{bmatrix}=8sw+4tx+2uv-4uw-8tv-2sx=3$

According to the sum property, we have:

$\begin{gathered} det\begin{bmatrix}{32-s} & {16-t} & {64-u} \\ {v} & {w} & {x} \\ {4} & {2} & {8}\end{bmatrix}=det\begin{bmatrix}{32} & {16} & {64} \\ {0} & {0} & {0} \\ {0} & {0} & {0}\end{bmatrix}+det\begin{bmatrix}{-s} & {-t} & {-u} \\ {v} & {w} & {x} \\ {4} & {2} & {8}\end{bmatrix} \\ det\begin{bmatrix}{32} & {16} & {64} \\ {0} & {0} & {0} \\ {0} & {0} & {0}\end{bmatrix}=0 \\ det\begin{bmatrix}{-s} & {-t} & {-u} \\ {v} & {w} & {x} \\ {4} & {2} & {8}\end{bmatrix}=-det\begin{bmatrix}{s} & {t} & {u} \\ {v} & {w} & {x} \\ {4} & {2} & {8}\end{bmatrix}=-3 \\ \therefore det\begin{bmatrix}32{-s} & {16-t} & {64-u} \\ {v} & {w} & {x} \\ {4} & {2} & {8}\end{bmatrix}=-3 \end{gathered}$

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