Linear Algebra Evaluating Determinants Problems 2

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:

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

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

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

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

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

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

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