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Assume that [tex][tex]$A$[/tex][/tex] and [tex][tex]$B$[/tex][/tex] are non-singular matrices of the same order. Solve for [tex][tex]$X$[/tex][/tex] in the matrix equation below:
[tex]\[ \left(B\left(X^t-I\right)^t A\right)^t + B = A^t \][/tex]
Certainly! Let's carefully solve for [tex]\( X \)[/tex] in the given matrix equation: [tex]\[
\left(B\left(X^t - I\right)^t A\right)^t + B = A^t
\][/tex]
We will solve this step-by-step.
### Step 1: Transpose the entire equation First, we need to eliminate the outer transpose by transposing the entire equation: [tex]\[
\left( \left(B\left(X^t - I\right)^t A\right)^t \right)^t + B^t = \left(A^t\right)^t
\][/tex]
Since transposing twice returns the original matrix, we have: [tex]\[
B\left(X^t - I\right)^t A + B^t = A
\][/tex]
### Step 2: Rearrange the term involving [tex]\( B \)[/tex] We need to simplify the equation: [tex]\[
B\left(X^t - I\right)^t A + B = A^t
\][/tex]
### Step 3: Subtract [tex]\( B \)[/tex] from both sides Move [tex]\( B \)[/tex] to the right-hand side of the equation: [tex]\[
B\left(X^t - I\right)^t A = A^t - B
\][/tex]
### Step 4: Isolate the term involving [tex]\( X \)[/tex] Next, we need to deal with the term [tex]\(\left(X^t - I\right)^t\)[/tex]. As [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are non-singular, they have inverses [tex]\( A^{-1} \)[/tex] and [tex]\( B^{-1} \)[/tex]. We will use these inverses to isolate the term [tex]\(\left(X^t - I\right)^t\)[/tex]:
Multiply both sides by [tex]\( B^{-1} \)[/tex] from the left: [tex]\[
\left(X^t - I\right)^t A = B^{-1}(A^t - B)
\][/tex]
Next, multiply both sides by [tex]\( A^{-1} \)[/tex] from the right: [tex]\[
\left(X^t - I\right)^t = B^{-1}(A^t - B) A^{-1}
\][/tex]
### Step 5: Transpose to return to the [tex]\( X \)[/tex] form To return to the original form of [tex]\( X \)[/tex], we need to transpose both sides of the equation: [tex]\[
X^t - I = \left( B^{-1}(A^t - B) A^{-1} \right)^t
\][/tex]
### Step 6: Solve for [tex]\( X^t \)[/tex] Next, we need to isolate [tex]\( X^t \)[/tex]:
[tex]\[
X^t = \left( B^{-1}(A^t - B) A^{-1} \right)^t + I
\][/tex]
### Step 7: Transpose to solve for [tex]\( X \)[/tex] Finally, we transpose the equation to solve for [tex]\( X \)[/tex]:
[tex]\[
X = \left( \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \right)^t
\][/tex]
### Final Solution Hence, the solution for [tex]\( X \)[/tex] is: [tex]\[
X = \left( \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \right)^t
\][/tex]
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