Question

If \(A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\) and \(B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}\), then find the value of \((2A-B)\).

Hint:

Compute \(2A\) entrywise, then subtract \(B\) entrywise.

Answer 1

Step 1: Combine first, then scale each pair:
Compute each entry directly as \(2a_{ij}-b_{ij}\) in one step rather than forming \(2A\) first: e.g. entry (1,1): \(2(1)-3=-1\); (1,2): \(2(2)-(-1)=5\); (1,3): \(2(3)-3=3\).

Step 2: Second row similarly:
(2,1): \(2(2)-(-1)=5\); (2,2): \(2(3)-0=6\); (2,3): \(2(1)-2=0\).

Final Answer:
Assembling gives \(\boxed{\begin{bmatrix}-1&5&3\\5&6&0\end{bmatrix}}\), matching the first method.