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Determinants

Example 9

Find minors and cofactors of all the elements of the determinant \(|\begin{smallmatrix}1&-2\\ 4&3\end{smallmatrix}|\)

Solution

Minor of the element \(a_{ij}\) is \(M_{ij}\) Here \(a_{11}=1\) So \(M_{11}=\) Minor of \(a_{11}=3\) \(M_{12}=\) Minor of the element \(a_{12}=4\) \(M_{21}=\) Minor of the element \(a_{21}=-2\) \(M_{22}=\) Minor of the element \(a_{22}=1\) Now, cofactor of \(a_{ij}\) is \(A_{ij}\) So \(A_{11}=(-1)^{1+1} M_{11}=(-1)^{2}(3)=3\) \(A_{12}=(-1)^{1+2} M_{12}=(-1)^{3}(4)=-4\) \(A_{21}=(-1)^{2+1} M_{21}=(-1)^{3}(-2)=2\) \(A_{22}=(-1)^{2+2} M_{22}=(-1)^{4}(1)=1\)