Question

Given matrices \( A \) and \( B \) where:

\[ A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} x & y \\ 1 & 2 \end{bmatrix} \]

and the condition:

\[ (A + B)(A - B) = A^2 - B^2 \]

If matrix \( C \) is defined as:

\[ C = \begin{bmatrix} x & 2 \\ 1 & y \end{bmatrix} \]

then the trace of \( C \) is:

\[ \text{Tr}(C) = x + y \]

• A. 3
• B. 5
• C. 7
• D. 9

Hint:

When working with matrices, remember that operations such as addition, subtraction, and multiplication must adhere to specific rules. The trace function, being a sum of diagonal elements, can sometimes simplify problems significantly.

Answer 1

The Correct Option is A

The trace of a matrix is the sum of its diagonal elements. For the matrix \( C = \begin{bmatrix} x & 2 \\ 1 & y \end{bmatrix} \), the trace is: \[ \text{Trace}(C) = x + y \] To find \(x\) and \(y\), we use the given matrix equation \( (A + B)(A - B) = A^2 - B^2 \). Simplifying each side of the equation using the properties of matrix addition and subtraction and then equating the resulting matrices: \[ A + B = \begin{bmatrix} 1+x & 2+y \\ 3 & 3 \end{bmatrix}, \quad A - B = \begin{bmatrix} 1-x & 2-y \\ 1 & -1 \end{bmatrix} \] \[ A^2 = \begin{bmatrix} 5 & 4 \\ 4 & 5 \end{bmatrix}, \quad B^2 = \begin{bmatrix} x^2+y & 2x+2y \\ x+2 & 5 \end{bmatrix} \] Equating \(A^2 - B^2\) and \((A + B)(A - B)\), solving for \(x\) and \(y\), we find that \(x = 1\) and \(y = 2\). Substituting back into the trace formula: \[ \text{Trace}(C) = 1 + 2 = 3 \] Thus, the trace of matrix \( C \) is 3, matching option (1).