Question

The 2×2 matrices P and Q satisfy the following relations: 

The matrix Q is equal to _______.

• A. A
• B. B
• C. C
• D. D

Hint:

To solve for a matrix when given the sum and difference of matrices, add or subtract the equations and then divide by 2 to isolate the matrix of interest.

Answer 1

The Correct Option is A

We are given the equations \( P + Q \) and \( P - Q \). To find \( Q \), we can add these two equations:

\[ (P + Q) + (P - Q) = \begin{pmatrix} 3 & 1 \\ 2 & 12 \end{pmatrix} + \begin{pmatrix} -1 & -7 \\ 8 & 2 \end{pmatrix}. \]

This simplifies to:

\[ 2P = \begin{pmatrix} 2 & -6 \\ 10 & 14 \end{pmatrix}. \]

Thus,

\[ P = \frac{1}{2} \begin{pmatrix} 2 & -6 \\ 10 & 14 \end{pmatrix} = \begin{pmatrix} 1 & -3 \\ 5 & 7 \end{pmatrix}. \]

Now, subtract \( P - Q \) from \( P + Q \):

\[ (P + Q) - (P - Q) = \begin{pmatrix} 3 & 1 \\ 2 & 12 \end{pmatrix} - \begin{pmatrix} -1 & -7 \\ 8 & 2 \end{pmatrix}. \]

This gives:

\[ 2Q = \begin{pmatrix} 4 & 8 \\ -6 & 10 \end{pmatrix}, \]

so

\[ Q = \frac{1}{2} \begin{pmatrix} 4 & 8 \\ -6 & 10 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ -3 & 5 \end{pmatrix}. \]

Thus, the correct answer is (A).

Final Answer:

\[ \boxed{(A)\, \begin{pmatrix} 2 & 4 \\ -3 & 5 \end{pmatrix}.} \]