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Evaluate the determinant for each one of the following matrices:

$(i)\left(\begin{array}{lll}{1} & {0} & {1} \\ {0} & {4} & {1} \\ {0} & {0} & {5}\end{array}\right)$

$\Rightarrow det(A)=1 \times 4 \times 5=20$

$\left(\begin{array}{ccc}{+} & {-} & {+} \\ {-} & {+} & {-} \\ {+} & {-} & {+}\end{array}\right) \Rightarrow \text {det}(A)=1\left|\begin{array}{cc}{4} & {1} \\ {0} & {5}\end{array}\right|=1 \times(4 \times 5)-0=20$

$(i i)\left(\begin{array}{llll}{2} & {0} & {2} & {1} \\ {0} & {4} & {4} & {1} \\ {0} & {0} & {2} & {7} \\ {0} & {0} & {0} & {5}\end{array}\right)$

$\Rightarrow d e t(B)=2 \times 4 \times 2 \times 5=80$

Solve the equation:

$\left|\begin{array}{ccc}{1} & {x} & {x^{2}} \\ {1} & {1} & {1} \\ {1} & {-3} & {9}\end{array}\right|=0$

$\left(\begin{array}{ccc}{+} & {-} & {+} \\ {-} & {+} & {-} \\ {+} & {-} & {+}\end{array}\right)$

$\Rightarrow\left|\begin{array}{cc}{1} & {1} \\ {-3} & {9}\end{array}\right|-x\left|\begin{array}{cc}{1} & {1} \\ {1} & {9}\end{array}\right|+x^{2}\left|\begin{array}{cc}{1} & {1} \\ {1} & {-3}\end{array}\right|=0$

$=(9-(-3))-x(9-1)+x^{2}(-3-1)=0$

$=12-8 x-4 x^{2}=0$

$=x^{2}+2 x-3=0$

$(x+3)(x-1)=0 \Rightarrow x=-3, x=1$