Question

5. (a) Prove that det(AAT ) = (det(A))2.

(b) Suppose that A is an n×n matrix such that AT = −A. (Such an A
is called a skew- symmetric matrix.) If n is odd, prove that det(A)
= 0.

Answer #1

1. If A is an n n matrix, prove that
(a) ATA is a symmetric matrix.
(b) A + AT is a symmetric matrix and A -
AT is a skew-symmetric matrix.
(c) A is the sum of a symmetric and a skew-symmetric matrix.

If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A
and B are 2x2 matrices, can I use that to show that Det(A)Det(B) =
Det(AB) for any n x n matrix? If so how?

Let A be an n × n-matrix. Show that there exist B, C such that B
is symmetric, C is skew-symmetric, and A = B + C. (Recall: C is
called skew-symmetric if C + C^T = 0.) Remark: Someone answered
this question but I don't know if it's right so please don't copy
his solution

Let A be an n×n nonsingular matrix. Denote by adj(A) the
adjugate matrix
of A. Prove:
1) det(adj(A)) = (det(A))
2) adj(adj(A)) = (det(A))n−2A

prove that if E is elementary matrix, then E^T is also elementary
matrix and det(E^T)=det(E)

n×n-matrix M is symmetric if M = M^t. Matrix M is
anti-symmetric if M^t = -M.
1. Show that the diagonal of an anti-symmetric matrix are
zero
2. suppose that A,B are symmetric n × n-matrices. Prove that AB
is symmetric if AB = BA.
3. Let A be any n×n-matrix. Prove that A+A^t is symmetric and A
- A^t antisymmetric.
4. Prove that every n × n-matrix can be written as the sum of a
symmetric and anti-symmetric matrix.

Suppose that M is an n×n matrix of finite order. Find all
possible values for det(M).

Prove: If A is an n × n symmetric matrix all of whose
eigenvalues are nonnegative, then xTAx ≥ 0 for all
nonzero x in the vector space Rn.

(3 pts) Let A be a square n × n matrix whose rows are
orthogonal. Prove that the columns of A are also orthogonal.
Hint: The orthogonality of rows is equivalent to AAT
= I ⇒ ATAAT = AT

A matrix A is called orthonormal if AAT = I. (a) Show that an
orthonormal matrix is invertible and that the inverse is
orthonormal. (b)
Showtheproductoftwoorthonormalmatricesisalsoorthonormal. (c) By
trials and errors, nd three orthonormal matrices of order 2. (d)
Let x be a real number, show that the matrices A =cosx −sinx sinx
cosx, B = cosx sinx −sinx cosx are orthonormal.

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