Characteristic values of the upper triangular matrix A=( ...

Characteristic values of the upper triangular matrix $\mathrm{A}=\left(\begin{array}{ccc}5 & 4 & -7 \\ 0 & -3 & 1 \\ 0 & 0 & 6\end{array}\right)$ are
(a) $2,5,1$
(b) $5,-3,6$
(c) $5,4,-1$
(d) $-7,-3,0$

Key Concepts

Characteristic Polynomial

The characteristic polynomial of a matrix is a polynomial in ? derived by computing the determinant of (A - ?I). Its roots are the eigenvalues of the matrix. This polynomial plays a central role in linear algebra, as it encapsulates essential spectral properties of the matrix.

Eigenvalues

Eigenvalues are scalars associated with a matrix that satisfy the equation A*v = ?*v, where v is a nonzero vector (the eigenvector). They are found as the roots of the characteristic polynomial obtained from det(A - ?I) = 0, and they provide deep insights into the matrix’s behavior in transformation and change of basis.

Triangular Matrix Properties

In the context of triangular matrices, the eigenvalues are immediately seen to be the entries on the main diagonal. This is due to the fact that the determinant of a triangular matrix equals the product of its diagonal entries, thus simplifying the calculation of the characteristic polynomial and the determination of eigenvalues.

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