Let A be a n Times n matrix answer the following questions W

Let A be a n Times n matrix: answer the following questions What is the condition of matrix A to have inverse? Discuss this in terms of rank, determinant, linear dependence of rows and columns of matrix A, singular or nonsingular. Show that if A has inverse then A^T also has inverse Homogenous linear system. Assume we have a homogenous system that has n equations and n unknowns, list a condition of this system to have non-trivial solution.

Solution

(a)

For A to have inverse:

Rank of the A should be n

Determinent of A is non zero

rows and columns of A should be linearly independent

A should be non-singular

(ii)

let A has a inverse A^-1

AA^-1 = I = A^-1 A

=>

(AA^-1 )^t= I =( A^-1 A)^t

=>
A^-1t A^t = I = A^tA^-1t

=>

A^-1 has a inverse

(b)

the matrix formed by the coefficients should be non-singular