Find minors and cofactors of the elements a11, a21 in the determinant \(\Delta=\left|\begin{array}{lll} {a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}} \end{array}\right|\)
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Find minors and cofactors of the elements a11, a21 in the determinant \(\Delta=\left|\begin{array}{lll} {a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}} \end{array}\right|\)
Answer
By definition of minors and cofactors, we have
Minor of a11 = M11 = \(\left|\begin{array}{ll} {a_{22}} & {a_{23}} \\ {a_{32}} & {a_{33}} \end{array}\right|\) = a22 a33 – a23 a32
Cofactor of a11 = A11 = (–1)1+1 M11 = a22 a33 – a23 a32
Minor of a21 = M21 = \(\left|\begin{array}{ll} {a_{12}} & {a_{13}} \\ {a_{32}} & {a_{33}} \end{array}\right|\) = a12 a33 – a13 a32
Cofactor of a21 = A21 = (–1)2+1 M21 = (–1) (a12 a33 – a13 a32) = – a12 a33 + a13 a32
Minor of a11 = M11 = \(\left|\begin{array}{ll} {a_{22}} & {a_{23}} \\ {a_{32}} & {a_{33}} \end{array}\right|\) = a22 a33 – a23 a32
Cofactor of a11 = A11 = (–1)1+1 M11 = a22 a33 – a23 a32
Minor of a21 = M21 = \(\left|\begin{array}{ll} {a_{12}} & {a_{13}} \\ {a_{32}} & {a_{33}} \end{array}\right|\) = a12 a33 – a13 a32
Cofactor of a21 = A21 = (–1)2+1 M21 = (–1) (a12 a33 – a13 a32) = – a12 a33 + a13 a32
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