91Ó°ÊÓ

Choose a row or column to expand along. For simplicity, expand along the first row.

Calculate the minors and cofactors for each element in the selected row or column. The determinant is then the sum of products, each product being the original matrix element times the corresponding cofactor. The term 'cofactor' itself includes the sign for each term, which is determined by the formula \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column numbers for the element being considered. Calculate these following this formulas: Cofactor of \(w\) = \((-1)^{1+1} \times\) \[\begin{vmatrix} 15 & -25 & 30 \ 20 & -15 & -10 \ 35 & -25 & -40 \end{vmatrix}\], Cofactor of \(x\) = \((-1)^{1+2} \times\) \[\begin{vmatrix} 10 & -25 & 30 \ -30 & -15 & -10 \ 30 & -25 & -40 \end{vmatrix}\], Cofactor of \(y\) = \((-1)^{1+3} \times\) \[\begin{vmatrix} 10 & 15 & 30 \ -30 & 20 & -10 \ 30 & 35 & -40 \end{vmatrix}\], Cofactor of \(z\) = \((-1)^{1+4} \times\) \[\begin{vmatrix} 10 & 15 & -25 \ -30 & 20 & -15 \ 30 & 35 & -25 \end{vmatrix}\].

These are 3x3 determinants, which can be solved in a similar fashion. Once these are calculated, the results are plugged into the following determinant formula for the first row: \[w \times \text{{cofactor of }} w - x \times \text{{cofactor of }} x + y \times \text{{cofactor of }} y - z \times \text{{cofactor of }} z.\]