Question

If \(A=\begin{pmatrix}6 & 2\\7 & -5\end{pmatrix}\) and \(A-B=\begin{pmatrix}-2 & 1\\4 & -9\end{pmatrix}\) then \(B=\begin{pmatrix}8 & 1\\3 & 4\end{pmatrix}\)

• A. $\begin{pmatrix}-8&-1
  3&4\end{pmatrix}$
• B. $\begin{pmatrix}8&1
  -3&-4\end{pmatrix}$
• C. $\begin{pmatrix}4&3
  11&-14\end{pmatrix}$
• D. $\begin{pmatrix}8&1
  3&4\end{pmatrix}$
• E. $\begin{pmatrix}-3&4
  5&-3\end{pmatrix}$

Hint:

Matrices Tip: Always be careful with double negatives when subtracting matrices containing negative numbers. Writing out the intermediate step like $6 - (-2)$ prevents careless errors.

Answer 1

The Correct Option is D

Concept:
Matrix addition and subtraction operate element-by-element, just like standard algebraic equations. If you are given matrix $A$ and the result of $A - B = C$, you can rearrange the equation algebraically to isolate $B$: $B = A - C$.

Step 1: Set up the algebraic relationship.
We are given: $$A - B = \begin{pmatrix}-2 & 1\\ 4 & -9\end{pmatrix}$$ To find $B$, rearrange: $$B = A - \begin{pmatrix}-2 & 1\\ 4 & -9\end{pmatrix}$$

Step 2: Substitute Matrix A into the equation.
$$B = \begin{pmatrix}6 & 2\\ 7 & -5\end{pmatrix} - \begin{pmatrix}-2 & 1\\ 4 & -9\end{pmatrix}$$

Step 3: Element-wise subtraction.
$$B = \begin{pmatrix}6 - (-2) & 2 - 1\\ 7 - 4 & -5 - (-9)\end{pmatrix}$$

Step 4: Simplify signs.
$$B = \begin{pmatrix}6 + 2 & 2 - 1\\ 7 - 4 & -5 + 9\end{pmatrix}$$

Step 5: Final computation.
$$B = \begin{pmatrix}8 & 1\\ 3 & 4\end{pmatrix}$$ Hence the correct answer is (D) $\begin{pmatrix}8 & 1\\ 3 & 4\end{pmatrix}$.