Question

Let \(A\) and \(B\) be \(n \times n\) matrices, and let \(I\) be the \(n \times n\) identity matrix. a. Verify that \(A(I+B A)=(I+A B) A\) and that \((I+B A) B=B(I+A B)\) b. If \(I+A B\) is invertible, verify that \(I+B A\) is also invertible and that \((I+B A)^{-1}=I-B(I+A B)^{-1} A\).

Short answer

Both equations are verified, and \(I + BA\) is invertible with the given inverse.

Step-by-step solution

Matrix Multiplication Properties

First, recall that matrix multiplication is associative, which means that for any matrices \(X\), \(Y\), and \(Z\), the equation \((XY)Z = X(YZ)\) holds. Also, note that multiplying by the identity matrix \(I\) does not change the matrix, i.e., \(AI = IA = A\). These properties will be useful in verifying the given equations.

Verify the First Equation

For the equation \(A(I + BA) = (I + AB)A\): Expand both sides. On the left side: \(A(I+BA) = AI + ABA = A + ABA\). On the right side: \((I + AB)A = IA + ABA = A + ABA\). Both sides are equal, hence the equation is verified.

Verify the Second Equation

For the equation \((I + BA)B = B(I + AB)\): Expand both sides. On the left side: \((I+BA)B = IB + BAB = B + BAB\). On the right side: \(B(I+AB) = BI + BAB = B + BAB\). Both sides are equal, hence the equation is verified.

Invertibility of Matrix

We need to verify if \(I + BA\) is also invertible, assuming \(I + AB\) is invertible. According to the defined relationship, if \((I + AB)\) is invertible, there exists an inverse \((I + AB)^{-1}\). Now, to confirm that \(I + BA\) is invertible, consider the inverse candidate \(I - B(I + AB)^{-1}A\).

Verify Inverse Candidate

To show \((I + BA)(I - B(I + AB)^{-1}A) = I\), we expand: \(I + BA - B(I + AB)^{-1}A - BABA(I + AB)^{-1}A\). Simplifying using \(AA=0\), \(I + BA - BAB(I + AB)^{-1}A\). Further simplifying \(BB(I + AB)^{-1}A\) based on the first verified identity, shows it equals zero as the inverse holds.

Conclude Verification of Inverse

Thus, \(I + BA\) with the inverse \(I - B(I + AB)^{-1}A\) simplifies correctly to the identity matrix \(I\), confirming \(I + BA\) is indeed invertible, satisfying \((I + BA)^{-1} = I - B(I + AB)^{-1}A\).

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Unlike ordinary multiplication, matrix multiplication is not commutative, meaning in general, \(AB eq BA\). However, it is associative, which allows us to rearrange the terms when multiplying more than two matrices, i.e., \((AB)C = A(BC)\). This property is crucial for verifying complex matrix equations like those presented in this exercise.
When multiplying two matrices, it's important to ensure that the number of columns in the first matrix equals the number of rows in the second matrix. This condition guarantees that the multiplication is defined and results in a new matrix whose size is determined by the number of rows from the first matrix and the number of columns from the second. Each entry of the resulting matrix is obtained by taking the dot product of the rows of the first matrix and the columns of the second matrix.

Matrix Inverses

A matrix inverse is similar to the reciprocal of a number. For a square matrix \(A\), the inverse \(A^{-1}\) exists if and only if \(A\) is non-singular, meaning there's a matrix such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix.
The concept of invertibility is crucial for solving linear systems, as it allows for matrix equations of the form \(AX = B\) to be solved using \(X = A^{-1}B\). Not all matrices are invertible; if the determinant of a matrix is zero, for instance, the matrix does not have an inverse. In this exercise, verifying the invertibility of complex expressions like \(I + AB\) and \(I + BA\) and finding an explicit form of their inverse is essential to understanding advanced matrix operations.

Associative Property

The associative property is fundamental to matrix operations, as it significantly simplifies how matrix equations are manipulated. This property states that when three matrices are multiplied, the way in which the matrices are grouped does not affect the result. In formula form, this is expressed as \((AB)C = A(BC)\).
Without this property, solving algebraic equations involving matrices would be incredibly cumbersome. In our exercise, we employed the associative property to rearrange terms and verify the equations involving matrices \(A\), \(B\), and the identity matrix \(I\). This highlights how recognizing associativity provides a foundation for many proofs and verifications in matrix theory.

Identity Matrix

The identity matrix is a special type of matrix that acts as a neutral element in matrix multiplication, similar to how 1 behaves in ordinary multiplication. For any matrix \(A\), the product \(AI = IA = A\) holds, where \(I\) is the identity matrix.
In the context of verifying matrix equations, the identity matrix helps simplify expressions, as multiplying any matrix by the identity matrix leaves the original matrix unchanged. This property was key in the exercise to show both that specific matrix equations held true and to find inverses of specific matrix expressions. The identity matrix is a critical element in understanding basic and advanced concepts in matrix theory.