Question

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

Answer #1

matrix A (nxn). Prove that the sum of the eigenvalues of a
matrix A equals to the sum of its diagonal elements (Aii) using the
similarity of transformation's notation.

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:
Why must every upper triangular matrix with no zero entries on
the main diagonal be nonsingular?
(Linear Algebra)

Prove that every nonzero element of Zn is either a unit or a
zero divisor, but not both.

Prove that every nonzero coefficient of the Taylor series of (1
- x + x²) eˣ

Let R*= R\ {0} be the set of nonzero real
numbers. Let
G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
R*, b in R}
(a) Prove that G is a subgroup of GL(2,R)
(b) Prove that G is Abelian

Prove that if a is a transcendental number, then a^n is also
transcendental for all nonzero integers n.

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.

Show/Prove that every invertible square (2x2) matrix is a
product of at most four elementary matrices

1. Decide whether each statement is true or false. Prove your
answer (i.e. prove that it is true or prove that it is false.)
(a) There exists a nonzero integer α such that α · β is an
integer for every rational number β.
(b) For every rational number β, there exists a nonzero integer
α such that α · β is an integer.

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