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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:
$$\left|\begin{array}{ccc}{1} & {x} & {x^{2}} \\ {1} & {1} & {1} \\ {1} & {-3} & {9}\end{array}\right|=0$$
$$\left(\begin{array}{ccc}{+} & {-} & {+} \\ {-} & {+} & {-} \\ {+} & {-} & {+}\end{array}\right)$$
$$\Rightarrow\left|\begin{array}{cc}{1} & {1} \\ {-3} & {9}\end{array}\right|-x\left|\begin{array}{cc}{1} & {1} \\ {1} & {9}\end{array}\right|+x^{2}\left|\begin{array}{cc}{1} & {1} \\ {1} & {-3}\end{array}\right|=0$$
$$=(9-(-3))-x(9-1)+x^{2}(-3-1)=0$$
$$=12-8 x-4 x^{2}=0$$
$$=x^{2}+2 x-3=0$$
$$(x+3)(x-1)=0 \Rightarrow x=-3, x=1$$
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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:
$$\left|\begin{array}{ccc}{1} & {x} & {x^{2}} \\ {1} & {1} & {1} \\ {1} & {-3} & {9}\end{array}\right|=0$$
$$\left(\begin{array}{ccc}{+} & {-} & {+} \\ {-} & {+} & {-} \\ {+} & {-} & {+}\end{array}\right)$$
$$\Rightarrow\left|\begin{array}{cc}{1} & {1} \\ {-3} & {9}\end{array}\right|-x\left|\begin{array}{cc}{1} & {1} \\ {1} & {9}\end{array}\right|+x^{2}\left|\begin{array}{cc}{1} & {1} \\ {1} & {-3}\end{array}\right|=0$$
$$=(9-(-3))-x(9-1)+x^{2}(-3-1)=0$$
$$=12-8 x-4 x^{2}=0$$
$$=x^{2}+2 x-3=0$$
$$(x+3)(x-1)=0 \Rightarrow x=-3, x=1$$
No comments yet
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