Question

If​ A, B, and C are n times nn×n invertible​ matrices, does the equation Upper C Superscript negative 1 Baseline (Upper A plus Upper X )Upper B Superscript negative 1 Baseline equals Upper I Subscript nC−1(A+X)B−1=In have a solution​ X? If​ so, find it.Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The solution is X=___B. There is no solution

Answer 1

Looks like the matrix equation is supposed to be

$C^{-1}(A+X)B^{-1}=I_n$

where $I_n$ presumably denotes the $n\times n$ identity matrix.

Since $A,B,C$ are all invertible, we have by multiplying on the left by $C$,

$C(C^{-1}(A+X)B^{-1})=CI_n$

$(CC^{-1})((A+X)B^{-1})=C$

$(A+X)B^{-1}=C$

then multiplying on the right by $B$,

$((A+X)B^{-1})B=CB$

$(A+X)(B^{-1}B)=CB$

$A+X=CB$

and finally subtracting $A$ from both sides to end up with

$X=CB-A$