Question

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

Answer #1

If A is a non singular n × n matrix show that:
a) adj(A) is non singular
b) [adj(A)]^−1 = det(A^−1 )A = adj(A^−1 )

Let A be an n×n matrix. If there exists k > n such that A^k
=0,then
(a) prove that In − A is nonsingular, where In is the n × n
identity matrix;
(b) show that there exists r ≤ n such that A^r= 0.

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.

Prove that if A is a nonsingular nxn matrix, then so is cA for
every nonzero real number c.

Let A be a (n × n) matrix. Show that A and AT have
the same characteristic polynomials (and therefore the same
eigenvalues). Hint: For any (n×n) matrix B, we have
det(BT) = det(B). Remark: Note that, however, it is
generally not the case that A and AT have the same
eigenvectors!

Show that for each nonsingular n x n matrix A there
exists a permutation matrix P such that P A has an LR
decomposition.
[note: A factorization of a matrix A into a product A=LR of a
lower (left) triangular matrix L and an upper (right) triangular
matrix R is called an LR decomposition of A.]

Let
A be an n by n matrix. Prove that if the linear transformation L_A
from F^n to F^n defined by L_A(v)=Av is invertible then A is
invertible.

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m
R^n is a linear transformation

Let ? be an eigenvalue of the ? × ? matrix A. Prove that ? + 1
is an eigenvalue of the matrix ? + ?? .

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