https://file.aiquickdraw.com/mathbot/file/1727401710230chv6m1ih.jpg

To calculate the determinant of the matrix in the image, we observe that the matrix is:

\[
\begin{pmatrix}
x & a & a & a \\
a & x & a & a \\
a & a & x & a \\
a & a & a & x
\end{pmatrix}
\]

This is a symmetric matrix with a special structure. The matrix elements are either \(x\) on the diagonal or \(a\) off the diagonal. Such matrices can often be simplified using properties of determinants.

The determinant of a matrix of this form can be calculated as:

\[
\det(A) = (x - a)^3 (x + 3a)
\]

This result comes from applying cofactor expansion and leveraging the symmetry of the matrix. Let's calculate the determinant for general values of \(x\) and \(a\):

\[
\det(A) = (x - a)^3 (x + 3a)
\]

Would you like a more detailed step-by-step solution or explanation? Here are five follow-up questions related to this matrix:

1. How does symmetry affect the determinant calculation of matrices like this?
2. Can you expand \((x - a)^3 (x + 3a)\) to see its simplified form?
3. What happens to the determinant if \(x = a\)?
4. What is the determinant of a similar matrix where \(a = 0\)?
5. How does the determinant behave as \(x \to 0\)?

**Tip:** For symmetric matrices, taking advantage of symmetry properties can simplify determinant calculations.

This problem involves calculating the determinant of a symmetric 4x4 matrix with diagonal and off-diagonal elements. Learn how to use cofactor expansion and the properties of symmetric matrices to simplify the computation. The formula det(A) = (x - a)^3 (x + 3a) provides a straightforward approach. Suitable for undergraduate students.

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