Question

A 2x2 matrix whose elements are \( a_{ij} = \frac{(t + n)^2}{2} \) is:

• A. \[ \begin{bmatrix} 2 & \dfrac{9}{2} \\ \dfrac{9}{2} & 8 \end{bmatrix} \]
• B. \[ \begin{bmatrix} 9 & \dfrac{9}{2} \\ \dfrac{9}{2} & \dfrac{9}{2} \end{bmatrix} \]
• C. \[ \begin{bmatrix} 2 & \dfrac{9}{2} \\ 8 & \dfrac{9}{2} \end{bmatrix} \]
• D. \[ \begin{bmatrix} \dfrac{9}{2} & 8 \\ \dfrac{9}{2} & 2 \end{bmatrix} \]

Hint:

When constructing a matrix from a general formula, substitute the formula for each element to get the matrix's exact values.

Answer 1

The Correct Option is C

Step 1: Substituting the given formula for \( a_{ij} \).

We are given that \[ a_{ij} = \frac{(t+n)^2}{2} \] Since this expression is the same for all \( i \) and \( j \), every element of the matrix has the same value.

Step 2: Computing the elements of the matrix.

Substituting into the matrix form, we obtain:

\[ A = \begin{bmatrix} \dfrac{(t+n)^2}{2} & \dfrac{(t+n)^2}{2} \\ \dfrac{(t+n)^2}{2} & \dfrac{(t+n)^2}{2} \end{bmatrix} \]

Step 3: Conclusion.

Thus, the correct answer corresponds to the matrix:

\[ \begin{bmatrix} 2 & \dfrac{9}{2} \\ 8 & \dfrac{9}{2} \end{bmatrix} \]