<p>If <span>\(A = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}\)</span> and <span>\(B = \begin{bmatrix} 1 & 4 \\ 7 & 2 \end{bmatrix}\)</span>, find <span>\(3A - 2B\)</span>.</p>
Step-by-Step Solution
Key Concept: Matrix operations (scalar multiplication and subtraction) are performed element-wise. First multiply each matrix by its scalar, then subtract corresponding elements.
<p><strong>Step 1:</strong> Calculate 3A by multiplying each element of A by 3:</p><p>3A = 3 × \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 9 & 3 \end{bmatrix}</p><p><strong>Step 2:</strong> Calculate 2B by multiplying each element of B by 2:</p><p>2B = 2 × \begin{bmatrix} 1 & 4 \\ 7 & 2 \end{bmatrix} = \begin{bmatrix} 2 & 8 \\ 14 & 4 \end{bmatrix}</p><p><strong>Step 3:</strong> Subtract 2B from 3A element-wise:</p><p>3A - 2B = \begin{bmatrix} 6 & -3 \\ 9 & 3 \end{bmatrix} - \begin{bmatrix} 2 & 8 \\ 14 & 4 \end{bmatrix}</p><p>= \begin{bmatrix} 6-2 & -3-8 \\ 9-14 & 3-4 \end{bmatrix}</p><p>= \begin{bmatrix} 4 & -11 \\ -5 & -1 \end{bmatrix}</p><p>∴ Answer: \begin{bmatrix} 4 & -11 \\ -5 & -1 \end{bmatrix}</p>
Correct Answer: \begin{bmatrix} 4 & -11 \\ -5 & -1 \end{bmatrix}