Solving Matrix Equations Find X In B(X⁻¹ - I)A + B = A - BX⁻¹

Jul 12, 2025 by ADMIN 62 views

Solve for X in Matrix Equation B(X⁻¹ - I)A + B = A - BX⁻¹

In the realm of matrix algebra, solving for an unknown matrix often requires a blend of algebraic manipulation and a deep understanding of matrix properties. This article delves into a specific matrix equation, providing a step-by-step solution to find the matrix X. We are given the equation:

B(X⁻¹ - I)A + B = A - BX⁻¹

where I represents the identity matrix, and the determinants of matrices A and B are non-zero, ensuring their invertibility. Let's embark on this mathematical journey to isolate X and discover its value.

Understanding the Problem

Before diving into the solution, it's crucial to understand the key components of the equation. We're dealing with matrices, which are rectangular arrays of numbers. Matrix operations, such as addition, subtraction, and multiplication, follow specific rules that differ from scalar arithmetic. The identity matrix, denoted by I, plays a role analogous to the number 1 in scalar algebra; when multiplied with any matrix, it leaves the matrix unchanged. The inverse of a matrix, denoted by X⁻¹, is a matrix that, when multiplied with the original matrix X, results in the identity matrix. The condition that det(A) ≠ 0 and det(B) ≠ 0 ensures that both matrices A and B have inverses, which is critical for our solution.

The matrix equation we are tasked with solving involves a combination of these operations. Our primary goal is to isolate the unknown matrix X. To achieve this, we will employ a series of algebraic manipulations, leveraging the properties of matrix operations and inverses.

Step-by-Step Solution

Expand the equation:

Our initial step involves expanding the left side of the equation to separate the terms. Distributing B across the parenthesis, we get:
```
BX⁻¹A - BIA + B = A - BX⁻¹
```
Since BI = B, we can simplify the equation to:
```
BX⁻¹A - BA + B = A - BX⁻¹
```
Rearrange the terms:

Next, we want to gather all terms containing X⁻¹ on one side of the equation. To achieve this, add BX⁻¹ to both sides:
```
BX⁻¹A - BA + B + BX⁻¹ = A
```
Now, move the terms without X⁻¹ to the right side by adding BA and subtracting B from both sides:
```
BX⁻¹A + BX⁻¹ = A + BA - B
```
Factor out X⁻¹:

We can factor out X⁻¹ from the left side of the equation. Notice that X⁻¹ is multiplied on the left in both terms, so we factor it out on the left:
```
B(X⁻¹A) + B(X⁻¹) = BX⁻¹A + BX⁻¹
```
However, we need to factor out X⁻¹ as a single term. To do this correctly, we factor X⁻¹ on the right:
```
(BA + B)X⁻¹ = BX⁻¹A + BX⁻¹
```
This step is crucial as matrix multiplication is not commutative (AB ≠ BA in general). Factoring out X⁻¹ on the right gives us:
```
B(A + I) = BX⁻¹(A + I)
```
The correct factoring should be:
```
BX⁻¹A + BX⁻¹ = B(A+I)X⁻¹
```
So, the equation becomes:
```
B(A + I)X⁻¹ = A + BA - B
```
Isolate X⁻¹:

To isolate X⁻¹, we need to eliminate the matrix B on the left side. Since det(B) ≠ 0, B has an inverse, B⁻¹. Multiply both sides of the equation on the left by B⁻¹:
```
B⁻¹[B(A + I)X⁻¹] = B⁻¹(A + BA - B)
```
Using the associative property of matrix multiplication, we have:
```
(B⁻¹B)(A + I)X⁻¹ = B⁻¹(A + BA - B)
```
Since B⁻¹B = I, the identity matrix, we get:
```
I(A + I)X⁻¹ = B⁻¹(A + BA - B)
```
Which simplifies to:
```
(A + I)X⁻¹ = B⁻¹(A + BA - B)
```
Now, we need to eliminate (A + I) from the left side. Let's assume that (A + I) is invertible. We multiply both sides on the left by (A + I)⁻¹:
```
(A + I)⁻¹(A + I)X⁻¹ = (A + I)⁻¹B⁻¹(A + BA - B)
```
This simplifies to:
```
X⁻¹ = (A + I)⁻¹B⁻¹(A + BA - B)
```
Solve for X:

To find X, we need to take the inverse of both sides of the equation. Recall that (X⁻¹)⁻¹ = X. Also, remember that the inverse of a product of matrices is the product of the inverses in reverse order. Therefore:
```
X = [(A + I)⁻¹B⁻¹(A + BA - B)]⁻¹
```
Applying the inverse property, we reverse the order and take the inverse of each term:
```
 X = (A + BA - B)⁻¹(B⁻¹)⁻¹((A + I)⁻¹)⁻¹
```
Since (B⁻¹)⁻¹ = B and ((A + I)⁻¹)⁻¹ = A + I, the equation simplifies to:
```
X = (A + BA - B)⁻¹B(A + I)
```

Final Result and Discussion

Therefore, the solution for X in the given matrix equation is:

 X = (A + BA - B)⁻¹B(A + I)

This result highlights the power of matrix algebra in solving complex equations. By carefully applying the rules of matrix operations and inverses, we were able to isolate the unknown matrix X. The solution involves several matrix multiplications and inversions, emphasizing the importance of understanding these operations. To ensure the existence of a unique solution, we relied on the given conditions that det(A) ≠ 0 and det(B) ≠ 0, which guarantee the invertibility of A and B. The invertibility of (A+I) and (A + BA - B) is also implicitly assumed for this solution to be valid. The process required careful attention to the order of operations, as matrix multiplication is not commutative. This step-by-step solution demonstrates a systematic approach to tackling matrix equations, providing a valuable framework for solving similar problems.

Key Takeaways

Solving matrix equations often involves algebraic manipulation similar to scalar equations, but with careful consideration of matrix properties.
The order of matrix multiplication is crucial, as it is not commutative.
The inverse of a matrix plays a vital role in isolating the unknown matrix.
The determinant of a matrix determines its invertibility.
Understanding the properties of the identity matrix is essential for simplifying matrix equations.
This problem exemplifies the importance of a systematic approach to solving complex mathematical problems.

By following these steps and understanding the underlying principles of matrix algebra, you can confidently tackle similar matrix equations and unravel the mysteries of linear transformations.