Consider the matrix
wherep, q, r, s are arbitrary real numbers. Compare this with Exercise 5.3.64.
a. Compute
b. For which values of p, q, r, s is A invertible? Find the inverse if it exists.
c. Find the determinant of A.
d. Find the complex eigenvalues of A.
e. If
f. Consider the numbers
and
Express the number
2183
as the sum of the squares of four integers:
Hint: Part e is useful. Note that
g. The French mathematician Joseph−Louis Lagrange (1736−1813) showed that any prime number can be expressed as the sum of the squares of four integers. Using this fact and your work in part (f) as a guide, show that any positive integer can be expressed in this way.
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