Given A = [(2,-1)(2,0)], B = [(-3,2)(4,0)] and C = [(1,0)(0,2)], find the matrix X such that A + X = 2B + C.

Question

Given A = \(\begin{bmatrix}2&-1\\[0.3em]2&0\end{bmatrix},\) B = \(\begin{bmatrix}-3&2\\[0.3em]4&0\end{bmatrix}\)and C = \(\begin{bmatrix}1&0\\[0.3em]0&2\end{bmatrix},\) find the matrix X such that A + X = 2B + C.

(a) \(\begin{bmatrix}5&-7\\[0.3em]2&6\end{bmatrix}\)

(b) \(\begin{bmatrix}-7&2\\[0.3em]6&5\end{bmatrix}\)

(c) \(\begin{bmatrix}-7&5\\[0.3em]6&2\end{bmatrix}\)

(d) \(\begin{bmatrix}-7&6\\[0.3em]5&2\end{bmatrix}\)

Answer 1

(c) A +  X = 2B + C⇒ X = 2B + C - A

⇒ X = 2 \(\begin{bmatrix}-3&2\\[0.3em]4&0\end{bmatrix}\) + \(\begin{bmatrix}1&0\\[0.3em]0&2\end{bmatrix}-\) \(\begin{bmatrix}2&-1\\[0.3em]2&0\end{bmatrix}\)

= \(\begin{bmatrix}-6&4\\[0.3em]8&0\end{bmatrix}\) + \(\begin{bmatrix}1&0\\[0.3em]0&2\end{bmatrix}-\)\(\begin{bmatrix}2&-1\\[0.3em]2&0\end{bmatrix}\)

= \(\begin{bmatrix}-6+1-2&4+0-(-1)\\[0.3em]8+0-2&0+2-0\end{bmatrix}\)= \(\begin{bmatrix}-7&5\\[0.3em]6&2\end{bmatrix}\)