Find $A(B C)$ given that $A=\left[\begin{array}{rr}3 & 9 \\ -8 & 8 \\ 8 & 2\end{array}\right], B=\left[\begin{array}{rr}-7 & 9 \\ 2 & -9\end{array}\right]$, and $C=\left[\begin{array}{rr}3 & -4 \\ -5 & 1\end{array}\right]$. If an operation is not defined, state the reason Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. $A(B C)=\quad$ (Simplify your answers ) B. This matrix operation is not possible because the product of $A$ and $B C$ is not defined C. This matrix operation is not possible because product $B C$ is not defined
Solution
Step 1 ：Given matrices A, B, and C as follows: \(A = \begin{bmatrix} 3 & 9 \\ -8 & 8 \\ 8 & 2 \end{bmatrix}\), \(B = \begin{bmatrix} -7 & 9 \\ 2 & -9 \end{bmatrix}\), and \(C = \begin{bmatrix} 3 & -4 \\ -5 & 1 \end{bmatrix}\).
Step 2 ：First, we need to calculate the product of matrices B and C. The product of two matrices is defined if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, both matrices B and C are 2x2, so their product is defined.
Step 3 ：By multiplying matrices B and C, we get \(BC = \begin{bmatrix} -66 & 37 \\ 51 & -17 \end{bmatrix}\).
Step 4 ：Next, we multiply the resulting matrix BC by A. The product of A and BC is defined if the number of columns in BC is equal to the number of rows in A. In this case, BC is a 2x2 matrix and A is a 3x2 matrix, so their product is defined.
Step 5 ：By multiplying matrix A with BC, we get \(ABC = \begin{bmatrix} 261 & -42 \\ 936 & -432 \\ -426 & 262 \end{bmatrix}\).
Step 6 ：Final Answer: The product of matrices A, B, and C is \(\boxed{\begin{bmatrix} 261 & -42 \\ 936 & -432 \\ -426 & 262 \end{bmatrix}}\).
