compute the matrix product \[ -3\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 6 & 0 & 1 \end{array}\right]\left[\begin{array}{cccc} 6 & 3 & -2 & -4 \\ 0 & -1 & 3 & 5 \\ 8 & 0 & -1 & 7 \\ -3 & 9 & -6 & 3 \end{array}\right]\left[\begin{array}{cccc} -1 & 3 & 0 & 4 \\ 0 & 6 & 2 & -5 \\ 4 & -2 & 3 & -1 \\ 5 & -7 & 4 & -3 \end{array}\right] \]
Solution
Step 1 ：The problem is asking for the product of three matrices. The first step is to multiply the first two matrices, and then multiply the result by the third matrix. The multiplication of matrices is done element by element, and then summing the results. The multiplication of matrices is associative, so the order in which the matrices are multiplied does not matter. However, it is not commutative, so the order of the matrices does matter.
Step 2 ：Let's denote the matrices as A, B, and C respectively. Matrix A is \[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 6 & 0 & 1 \end{array}\], Matrix B is \[\begin{array}{cccc} 6 & 3 & -2 & -4 \\ 0 & -1 & 3 & 5 \\ 8 & 0 & -1 & 7 \\ -3 & 9 & -6 & 3 \end{array}\], and Matrix C is \[\begin{array}{cccc} -1 & 3 & 0 & 4 \\ 0 & 6 & 2 & -5 \\ 4 & -2 & 3 & -1 \\ 5 & -7 & 4 & -3 \end{array}\].
Step 3 ：First, we multiply Matrix A and Matrix B, then multiply the result by Matrix C. The result is \[\begin{array}{cccc} 102 & -204 & 48 & -69 \\ -111 & 141 & -81 & 39 \\ -69 & 69 & -75 & -36 \\ -648 & 738 & -522 & 414 \end{array}\].
Step 4 ：So, the final answer is \(\boxed{\begin{array}{cccc} 102 & -204 & 48 & -69 \\ -111 & 141 & -81 & 39 \\ -69 & 69 & -75 & -36 \\ -648 & 738 & -522 & 414 \end{array}}\).
