Question

Answer the following question:

Find minor and cofactor of elements of the determinant:

${\displaystyle \left|\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right|}$

Answer 1

Here, ${\displaystyle \left|\begin{array}{ccc}{\text{a}}_{11}& {\text{a}}_{12}& {\text{a}}_{13}\\ {\text{a}}_{21}& {\text{a}}_{22}& {\text{a}}_{23}\\ {\text{a}}_{31}& {\text{a}}_{32}& {\text{a}}_{33}\end{array}\right|=\left|\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right|}$

M₁₁ = ${\displaystyle \left|\begin{array}{cc}0& -2\\ 0& 3\end{array}\right|}$ = 0 − 0 = 0

∴ C₁₁ = (– 1)¹⁺¹ M₁₁ = 1(0) = 0

M₁₂ = ${\displaystyle \left|\begin{array}{cc}3& -2\\ 1& 3\end{array}\right|}$ = 9 + 2 = 11

∴ C₁₂ = (– 1)¹⁺² M₁₂ = (– 1)(11) = −11

M₁₃ = ${\displaystyle \left|\begin{array}{cc}3& 0\\ 1& 0\end{array}\right|}$ = 0 –  0 = 0

∴ C₁₃ = (– 1)¹⁺³ M₁₃ = 1(0) = 0

M₂₁ = ${\displaystyle \left|\begin{array}{cc}-1& 2\\ 0& 3\end{array}\right|}$ = −3 − 0 = −3

∴ C₂₁ = (–1)²⁺¹ M₂₁ = (–1)(−3) = 3

M₂₂ = ${\displaystyle \left|\begin{array}{cc}1& 2\\ 1& 3\end{array}\right|}$ = 3 – 2 = 1

∴ C₂₂ = (–1)²⁺² M₂₂ = 1(1) = 1

M₂₃ = ${\displaystyle \left|\begin{array}{cc}1& -1\\ 1& 0\end{array}\right|}$ = 0 + 1 =  1

∴ C₂₃ = (–1)²⁺³ M₂₃ = (–1)(1) = –1

M₃₁ = ${\displaystyle \left|\begin{array}{cc}-1& 2\\ 0& -2\end{array}\right|}$ = 2 – 0 = 2

∴ C₃₁ = (–1)³⁺¹ M₃₁ = 1(2) = 2

M₃₂ = ${\displaystyle \left|\begin{array}{cc}1& 2\\ 3& -2\end{array}\right|}$ = –2 − 6 = −8

∴ C₃₂ = (–1)³⁺² M₃₂ = (–1)(−8) = 8

M₃₃ = ${\displaystyle \left|\begin{array}{cc}1& -1\\ 3& 0\end{array}\right|}$ = 0 + 3 = 3

∴ C₃₃ = (–1)³⁺³ M₃₃ = 1(3) = 3