You are given that the determinant of the matrix A = \begin{matrix} a & b \\ c & d \\ \end{matrix} is equal to 5. Using this information and the property of determinants determine the value of \begin{matrix} 2a+3c & 2b+3d \\ a & b \\ \end{matrix}
2026-04-25 19:25:56.1777145156
Given the determinant determine the value of the matrix
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Using the linearity of the determinant function
\begin{align*} \begin{vmatrix} 2a+3c & 2b+3d \\ a & b \\ \end{vmatrix} & = \begin{vmatrix} 2a & 2b \\ a & b \\ \end{vmatrix} + \begin{vmatrix} 3c & 3d \\ a & b \\ \end{vmatrix}\\ &=0 + \begin{vmatrix} 3c & 3d \\ a & b \\ \end{vmatrix}\\ &=3\begin{vmatrix} c & d \\ a & b \\ \end{vmatrix}\\ &=-15. \end{align*} You should try to see which properties have been used for each step.