Question

Let A be an *n*x*n* matrix. Prove that A is
invertible if and only if rank(A) = *n*.

Answer #1

Let A be an nxn matrix. Show that if Rank(A) = n, then Ax = b
has a unique solution for any nx1 matrix b.

Let A be square matrix prove that A^2 = I if and only if
rank(I+A)+rank(I-A)=n

Let A,B be a m by n matrix, Prove that
|rank(A)-rank(B)|<=rank(A-B)

Let A be an m × n matrix, and Q be an n × n invertible
matrix.
(1) Show that R(A) = R(AQ), and use this result to show that
rank(AQ) = rank(A);
(2) Show that rank(AQ) = rank(A).

Let A, B ? Mn×n be invertible matrices. Prove the following
statement: Matrix A is similar to B if and only if there exist
matrices X, Y ? Mn×n so that A = XY and B = Y X.

Let K be a positive definite matrix. Prove that K is invertible,
and that K^(-1) is also positive definite.

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 invertible matrix. Show that A∗ is invertible, and
that (A∗ ) −1 = (A−1 ) ∗ .

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.

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

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