Find Value of x

[nycmath]

See attachment.

[Math Tutor]

The matrix is diagonal (everything off the main diagonal is 0), so the determinant is simply the product of the diagonal entries:

(x − 4)(x + 4)(x + 1) = 0

If you prefer to expand along the first row, you get the same thing:
(x − 4)·[(x + 4)(x + 1) − 0·0] − 0 + 0 = (x − 4)(x + 4)(x + 1).

A product is zero exactly when one of the factors is zero, so

x = 4, x = −4, x = −1.

[nycmath]

The matrix is diagonal (everything off the main diagonal is 0), so the determinant is simply the product of the diagonal entries:

(x − 4)(x + 4)(x + 1) = 0

If you prefer to expand along the first row, you get the same thing:
(x − 4)·[(x + 4)(x + 1) − 0·0] − 0 + 0 = (x − 4)(x + 4)(x + 1).

A product is zero exactly when one of the factors is zero, so

x = 4, x = −4, x = −1.

Wow! Thank you for your time.

[Eigenvalue]

Here is my work for the question: