Calculate the Determinant of a Symmetric 4x4 Matrix

Math Problem Statement

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

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.

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Linear Algebra
Matrix Theory
Determinants
Symmetry

det(A) = (x - a)^3 (x + 3a)

Determinant of Symmetric Matrices
Cofactor Expansion

Undergraduate Level

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