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

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

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

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

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

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

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

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

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

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

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

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

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

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

$=-8+16=8$