Find the minors and cofactors of the elements of the determinant `Delta = |{:(a_(11), a_(12), a_(13)), (a_(21), a_(22), a_(23)), (a_(31), a_(32), a_(33)):}|`

Explanation

Let `M_(ij)` denote the minor of `a_(ij) " in "Delta`.
Now, `a_(11)` accurs in the 1st row and 1st column. So, in order to find the minor of `a_(11)`, we delete the 1st row and 1st column of `Delta`. The minor `M_(11) " of"a_(11)" is given by" M_(11) = |{:(a_(21), a_(23)), (a_(32), a_(33)):}| = (a_(22)a_(33) - a_(32)a_(23)).`
Similarly, we have
`M_(12) = |{:(a_(21), a_(23)), (a_(31), a_(33)):}| = (a_(21)a_(33) - a_(31)a_(23)),`
`M_(13) = |{:(a_(21), a_(22)), (a_(31), a_(32)):}| = (a_(21)a_(32) - a_(31)a_(22)),`
`M_(21) = |{:(a_(12), a_(13)), (a_(32), a_(32)):}| = (a_(12)a_(33) - a_(32)a_(13)).`
Similarly, we may obtain the minor of each of the remaining elements.
Now, if we denote the cofactor of `a_(ij) "by C_(ij)` then
`C_(11) = (-1)^(1+1)*M_(11) = M_(11) = (a_(22)a_(33) - a_(32)a_(23)),`
`C_(12) = (-1)^(1+2)*M_(12) = -M_(12) = (a_(31)a_(23) - a_(21)a_(33)),`
`C_(13) = (-1)^(1+3)*M_(13) = M_(13) = (a_(21)a_(32) - a_(31)a_(22)),`
`C_(21) = (-1)^(2+1)*M_(21) =-M_(21) = (a_(32)a_(13) - a_(12)a_(33)).`
Similarly, the cofactor of each of the remaining elements of `Delta` can be determined.