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Matrix Symmetry, Matrix Multiplication and Skew-Symmetric Matrices

2. Compute the product by inspection.

a) 3 0 0 2 1 b) 2 0 0 4 -1 3 -3 0 0
0 -1 0 -4 1 0 -1 0 1 2 0 0 5 0
0 0 2 2 5 0 0 4 -5 1 -2 0 0 2

8. Use the given equation to determine by inspection whether the matrices on the left commute.

a) 1 -3 4 1 = 1 -5 b) 2 -1 3 2 = 4 3
-3 2 1 2 -10 1 -1 3 2 1 3 1

16. Let A be an n x n symmetric matrix.

a) Show that Ak is symmetric is k is any nonnegative integer. Do in a proof.
b) If p(x) is a polynomial, is p(A) necessarily symmetric? Explain. Do in a proof.

22. A square matrix A is called skew-symmetric if AT = -A. Prove:

a) If A is an invertible skew-symmetric, then A-1 is skew symmetric.
b) If A and B are skew-symmetric, then so are AT, A + B, A - B, and kA for any scalar k.
c) Every square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Hint: Note the identity A = ½ (A + AT) + ½ (A - AT).

30. Indicate whether the statement is always true or sometimes false. Justify each answer in proof.

a) If AAT is singular, then so is A.
b) If A + B is symmetric, then so are A and B.
c) If A is an n x n matrix and Ax = 0 has only the trivial solution, then so does ATx = 0.
d) If A2 is symmetric, then so is A.

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