Evaluate \(A(B – C)\)
Question:
Evaluate: \[ \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 2 & 3 \end{bmatrix} \left( \begin{bmatrix} 1 & 0 & 2 \\ 2 & 0 & 1 \end{bmatrix} – \begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 2 \end{bmatrix} \right) \]
Evaluate: \[ \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 2 & 3 \end{bmatrix} \left( \begin{bmatrix} 1 & 0 & 2 \\ 2 & 0 & 1 \end{bmatrix} – \begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 2 \end{bmatrix} \right) \]
Solution:
Step 1: Subtract matrices
\[ B – C = \begin{bmatrix} 1-0 & 0-1 & 2-2 \\ 2-1 & 0-0 & 1-2 \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & -1 \end{bmatrix} \]Step 2: Multiply with A
\[ = \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & -1 \end{bmatrix} \] \[ = \begin{bmatrix} 1(1)+(-1)(1) & 1(-1)+(-1)(0) & 1(0)+(-1)(-1) \\ 0(1)+2(1) & 0(-1)+2(0) & 0(0)+2(-1) \\ 2(1)+3(1) & 2(-1)+3(0) & 2(0)+3(-1) \end{bmatrix} \] \[ = \begin{bmatrix} 0 & -1 & 1 \\ 2 & 0 & -2 \\ 5 & -2 & -3 \end{bmatrix} \]Final Answer:
\[ \boxed{ \begin{bmatrix} 0 & -1 & 1 \\ 2 & 0 & -2 \\ 5 & -2 & -3 \end{bmatrix} } \]