Question

If \(2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}\), then \(X\) is ______.

• A. \(\begin{bmatrix} 2 & 5 \\ 5 & 3 \end{bmatrix}\)
• B. \(\begin{bmatrix} -1 & 3 \\ 2 & -1 \end{bmatrix}\)
• C. \(\begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}\)
• D. \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

Hint:

Rearrange to 2X = (RHS matrix) - (given matrix), subtract entry-wise, then divide the whole result by 2.

Answer 1

The Correct Option is C

Step 1: Isolate the term containing X.
Starting from \(2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}\), subtract the known matrix from both sides: \[2X = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]

Step 2: Subtract the matrices entry by entry. \[2X = \begin{bmatrix} 3-1 & 8-2 \\ 7-3 & 2-4 \end{bmatrix} = \begin{bmatrix} 2 & 6 \\ 4 & -2 \end{bmatrix}\]

Step 3: Divide every entry by 2 to get X. \[X = \begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}\]

Step 4: Match with the options.
This is exactly option (C). Option (A) is what you get if you mistakenly add instead of subtracting the matrices before dividing by 2. Option (B) has the sign of the top-left entry flipped, a common slip when subtracting in the wrong order for that one cell. Option (D) is just the identity matrix and does not satisfy the original equation at all.