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Evaluate the determinant for each one of the following matrices:

$$(i)(2)$$

$$\Rightarrow$$ Let $$A=(2) \Rightarrow$$ det $$(A)=2$$

$$(i i)\left(\begin{array}{cc}{1} & {2} \\ {-1} & {3}\end{array}\right)$$

Let $$B=\left(\begin{array}{cc}{1} & {2} \\ {-1} & {3}\end{array}\right)\Rightarrow$$

det$$(B)=\left|\begin{array}{ll}{1} & {2} \\ {-1} & {3}\end{array}\right|=|B|$$

det $$(D)=(1 \times 3)-(2 x-1)$$

$$=3-(-2)=5$$

$$\left(\begin{array}{ccc}{2} & {-1} & {3} \\ {1} & {0} & {4} \\ {2} & {-3} & {1}\end{array}\right)$$

Minor $$\Rightarrow M_{12}=\left|\begin{array}{ll}{1} & {4} \\ {2} & {1}\end{array}\right|=1-8=-7$$

Cofactor $$Rightarrow (-1)^{i+j} \cdot M_{i j} \Rightarrow c_{12}=(-1)^{1+2} \cdot M_{12} $$

$$c_{12}=-M_{12} $$

$$=-\left|\begin{array}{ll}{1} & {4} \\ {2} & {1}\end{array}\right|=-(-7)=7$$

$$A=\left(\begin{array}{ccc}{2} & {-1} & {3} \\ {1} & {0} & {4} \\ {2} & {-3} & {1}\end{array}\right) \Rightarrow det (A)=2 c_{11}+1 c_{21}+2 c_{31} $$

$$det(A)=1 C_{21}+0+4 C_{23} $$

$$(-1)^{i+j}=\left(\begin{array}{ccc}{+} & {-} & {+} \\ {-} & {+} & {-} \\ {+} & {-} & {+}\end{array}\right)$$

$$A=\left(\begin{array}{ccc}{2} & {-1} & {3} \\ {1} & {0} & {4} \\ {2} & {-3} & {1}\end{array}\right)$$

$$A=\left(\begin{array}{ccc}{2} & {-1} & {3} \\ {1} & {0} & {4} \\ {2} & {-3} & {1}\end{array}\right)\left(\begin{array}{ccc}{+} & {-} & {+} \\ {-} & {+} & {-} \\ {+} & {-} & {+}\end{array}\right)$$

$$\text {det}(A)=2\left|\begin{array}{cc}{0} & {4} \\ {-3} & {1}\end{array}\right|-1\left|\begin{array}{cc}{-1} & {3} \\ {-3} & {1}\end{array}\right|+2\left|\begin{array}{cc}{-1} & {3} \\ {0} & {4}\end{array}\right|$$

$$=2[(0 \times 1)-(4 \times-3)]-\left[\left(-1 \times {1}\right)-(3 \times -3)\right]+2[(-1\times 4)-(3 \times 0)]$$

$$=2(0-(-12))-(-1-(-9))+2(-4-0)$$

$$=24-8+(-8)=8$$

$$\Rightarrow \text {det}(A)=-1\left|\begin{array}{rr}{-1} & {3} \\ {-3} & {1}\end{array}\right|-4\left|\begin{array}{rr}{2} & {-1} \\ {2} & {-3}\end{array}\right|$$

$$=-(-1-(-9))-4(-6-(-2))$$

$$=-8+16=8$$

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