In Exercises 5154 solve for the matrix X Assume that all mat

In Exercises 51-54, solve for the matrix X. Assume that all matrices are n x n and invertible as needed.

Solution

Given: AX(D+BX)^(-1) = C:

Multiply both sides on the right by (D+BX):
AX(D+BX)^(-1) * (D+BX) = C * (D+BX)

AX ((D+BX)^(-1) (D+BX)) = C(D+BX)
AX * I = CD + CBX
AX = CD + CBX.

Subtract CBX on both sides
AX - CBX = (CD + CBX) - CBX

AX - CBX = CD + (CBX - CBX)
AX - CBX = CD + 0
factorise: (A - CB)X = CD.

Multiply both sides on the left by (A - CB)^(-1):
we get : (A - CB)^(-1) * (A - CB)X = (A - CB)^(-1) * CD
((A - CB)^(-1) * (A - CB)) X = (A - CB)^(-1) CD
X = (A - CB)^(-1) CD

X = (A - CB)^(-1) CD.