Given:
The matrix A and B is,
[tex]A=\begin{bmatrix}{5} & {3} & {0} \\ {3} & {-1} & {0} \\ {0} & {0} & {2}\end{bmatrix},B=\begin{bmatrix}{c} & {3} & {0} \\ {3} & {d} & {0} \\ {0} & {0} & {e}\end{bmatrix}[/tex]
Take the matrix multiplication,
[tex]AB=\begin{bmatrix}{5c+9} & {15+3d} & {0} \\ {3c-3} & {9-d} & {0} \\ {0} & {0} & {2e}\end{bmatrix}[/tex]
Now, use the given condition,
[tex]\begin{gathered} AB=14I \\ \begin{bmatrix}{5c+9} & {15+3d} & {0} \\ {3c-3} & {9-d} & {0} \\ {0} & {0} & {2e}\end{bmatrix}=\begin{bmatrix}{14} & {0} & {0} \\ {0} & {14} & {0} \\ {0} & {0} & {14}\end{bmatrix} \\ \Rightarrow5c+9=14,15+3d=0 \\ 3c-3=0,9-d=14 \\ 2e=14 \\ \text{Take,}5c+9=14 \\ 5c=5 \\ c=1 \\ 15+3d=0 \\ 3d=-15 \\ d=-5 \\ And,2e=14\Rightarrow e=7 \end{gathered}[/tex]
Answer: c = 1, d = -5, e=7
