(1) Suppose that the matrix A is diagonal: d1,..., An E R A = diag(^1,..., An), Prove that A is negative definite if and only if d; < 0 for all i. (2) Let A1, A2, ... , An be the leading principal minors of the matrix A. Prove that A is negative definite if and only if (-1)*Ar < 0, k= 1,2,...,n

An n x n matrix A is skew symmetric if A^(-1)=A^t. Prove that if A is invertible and skew-symmetric that A^(-1) is also skew symmetric (use the properties of inverses.) You need to show that (A^(-1))^t=-A^(-1) . Select the antecedent of the theorem you are proving, then prove the theorem in the box provided. If attaching a file, use only pdf format.

Suppose A, B are 3x3 matrices, det A = 6 , and det B=-5. Find the following determinants. (a): det (B-'A) (b): det (AT B) (c): det (3B)

Prove or give a counterexample: For any r in real numbers, with n greater than or equal to 1, and matrix A in R^(n x n), one has det(r*A) = r*det(A).