Evaluate Row × Matrix × Column
Question:
Evaluate: \[ [1\;\;2\;\;3] \begin{bmatrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix} \]
Evaluate: \[ [1\;\;2\;\;3] \begin{bmatrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix} \]
Solution:
Step 1: Multiply first two matrices
\[ [1\;\;2\;\;3] \begin{bmatrix} 1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix} \] \[ = [1(1)+2(2)+3(0),\; 1(0)+2(0)+3(1),\; 1(2)+2(1)+3(2)] \] \[ = [5,\; 3,\; 10] \]Step 2: Multiply with column matrix
\[ [5\;\;3\;\;10] \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix} \] \[ = 5(2) + 3(4) + 10(6) \] \[ = 10 + 12 + 60 = 82 \]Final Answer:
\[ \boxed{82} \]