If Δ =
| a b c |
| d e f |
| g h i |
and A, B, C, D, G are the cofactors of the elements a, b, c, d, g respectively, then what is bB + cC - dD - gG equal to?
0
We are given the determinant \( \Delta \) of a 3x3 matrix: \[ \Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}, \] and we are tasked with finding the value of \( bB + cC - dD - gG \), where \( A \), \( B \), \( C \), \( D \), and \( G \) are the cofactors of the elements \( a \), \( b \), \( c \), \( d \), and \( g \) in the matrix, respectively. By the definition of cofactors, the cofactor of an element is the determinant of the submatrix obtained by removing the row and column of that element, multiplied by \( (-1)^{i+j} \), where \( i \) and \( j \) are the row and column indices of the element. Now, the expression \( bB + cC - dD - gG \) involves the cofactors corresponding to elements in the second column and the first row and column of the matrix. However, the important observation here is that the matrix is a part of the determinant expansion along the first row and the first column. The sum of these terms involves the properties of the cofactors in the determinant expansion, and due to symmetry and cancellation of terms, we find that: \[ bB + cC - dD - gG = 0. \] Therefore, the value of \( bB + cC - dD - gG \) is equal to 0.
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