Solving a matrix equation

Question

Solve for the matrix equation: 

X - XA = D for x 

please show all work

Answer 1

Hi Bianca,

 

X - XA = D

 

XI - XA = D, where I is the n x n identity

 

Factor out X out front.

 

X(I - A) = D

 

Assuming I - A is invertible, right multiply each side of the last equation by its inverse.

 

X(I - A)(I - A)^-1 = D(I - A)^-1

 

(I - A)(I - A)^-1 = I 

 

So, XI = D(I - A)^-1

 

Or

 

X = D(I - A)^-1