Example Question - variable elements

Determinants of Matrices with Variable Elements

Since this image contains determinant calculations and operations on matrices with variables, I will provide the solution to find the determinant based on the visible matrices in the image. <p>\text{Let's calculate the determinant of the first visible matrix:}</p> <p>\begin{vmatrix} a-1 & 3 & 5 \\ -1 & 3-5 & 5 \\ 1 & 3 & 5 \end{vmatrix}</p> <p>\text{Expanding along the first column:}</p> <p>(a-1)\begin{vmatrix} 3-5 & 5 \\ 3 & 5 \end{vmatrix} -(-1)\begin{vmatrix} -1 & 5 \\ 1 & 5 \end{vmatrix} + 1\begin{vmatrix} -1 & 3-5 \\ 1 & 3 \end{vmatrix}</p> <p>=(a-1)((3-5)(5)-5(3)) -(-1)((-1)(5)-5(1)) + (1)((-1)(3)-3(3-5))</p> <p>=(a-1)(-10+15) -(-1)(-5) + (1)(-3+6)</p> <p>=(a-1)(5) -5 + 3</p> <p>=5a-5 -5 +3</p> <p>=5a-7</p> <p>\text{Without more context or additional operations indicated for the other matrices, the calculation of their determinants isn't completed in the image.}</p>