## Compute the eigenvalues of a given matrix

Find the eigenvalues of the following matrix.

Recall that the eigenvalues of a matrix are the roots of the polynomial . Now in this case, we have

Using the Cofactor Expansion Formula for the determinant along the 4th row, (Theorem 29 on page 439 of D&F, with ), we have that . Since , this simplifies to

Note that both of these minor matrices are diagonal, and recall that the determinant of an upper- or lower-triangular matrix is the product of the diagonal entries. (To see this, consider the Leibniz formula for the determinant: . If is upper (lower) triangular, then for every , if moves at least one element in , then it must move some element to a strictly larger (smaller) one, in which case vanishes since some factor is zero. What remains is merely .)

So we have . Clearly the roots of this polynomial, hence the eigenvalues of , are and .

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## Comments

I’m currently working through AA:DF also, just for the hell of it, and I think this project is great, keep up the good work!

Thanks!