Problem 11 - Advanced Calculus

Question

Let \(\mathbf{A}\) be an \(n \times n\) matrix and suppose that \(\mathbf{B}\) and \(\mathbf{B}^{\prime}\) are two \(n \times n\) matrices with the property that $$ \mathbf{A B}=\mathbf{I}_{n}=\mathbf{B}^{\prime} \mathbf{A} $$ Show that \(\mathbf{B}=\mathbf{B}^{\prime}\) by verifying that $$ \mathbf{B}=\mathbf{I}_{n} \mathbf{B}=\left(\mathbf{B}^{\prime} \mathbf{A}\right) \mathbf{B}=\mathbf{B}^{\prime}(\mathbf{A} \mathbf{B})=\mathbf{B}^{\prime} \mathbf{I}_{n}=\mathbf{B}^{\prime} $$

Step-by-Step Solution

Answer

By substituting \( \mathbf{A} \mathbf{B} \) and \( \mathbf{B}' \mathbf{A} \) with the identity matrix \( \mathbf{I}_{n} \) and applying multiplication properties, it is shown that \( \mathbf{B} = \mathbf{B}' \).

Step 1: Verify Matrices Multiplication with Identity Matrix

First, recognize that when any matrix is multiplied by the identity matrix of the same dimension, the result is the original matrix. That is, for an identity matrix \( \mathbf{I}_{n} \) and any matrix \( \mathbf{X} \), the equality \( \mathbf{X} \mathbf{I}_{n} = \mathbf{X} \) holds.

Step 2: Multiply \( \mathbf{B} \) by Identity Matrix

Given that \( \mathbf{A} \mathbf{B} = \mathbf{I}_{n} \) and \( \mathbf{B}' \mathbf{A} = \mathbf{I}_{n} \), we multiply matrix \( \mathbf{B} \) by the identity matrix to get \( \mathbf{B} = \mathbf{I}_{n} \mathbf{B} \).

Step 3: Substitute \( \mathbf{B}' \mathbf{A} \) for \( \mathbf{I}_{n} \)

Since \( \mathbf{B}' \mathbf{A} = \mathbf{I}_{n} \), we can substitute \( \mathbf{B}' \mathbf{A} \) into the previous equation to get \( \mathbf{B} = (\mathbf{B}' \mathbf{A}) \mathbf{B} \).

Step 4: Reassociate the Matrix Multiplication

Use the associative property of matrix multiplication to group \( \mathbf{A} \) and \( \mathbf{B} \) together: \( \mathbf{B} = \mathbf{B}'(\mathbf{A} \mathbf{B}) \).

Step 5: Substitute \( \mathbf{A} \mathbf{B} \) with \( \mathbf{I}_{n} \)

Replace \( \mathbf{A} \mathbf{B} \) with \( \mathbf{I}_{n} \) since \( \mathbf{A} \mathbf{B} = \mathbf{I}_{n} \), which leads to \( \mathbf{B} = \mathbf{B}' \mathbf{I}_{n} \).

Step 6: Multiply \( \mathbf{B}' \) by Identity Matrix

Finally, as multiplication by the identity matrix leaves a matrix unchanged, we obtain \( \mathbf{B} = \mathbf{B}' \), thereby showing \( \mathbf{B} \) and \( \mathbf{B}' \) are indeed equal.

Key Concepts

Associative Property of Matrices

The Associative Property of Matrices is a fundamental property that allows us to regroup the factors in a matrix multiplication without altering the product. Specifically, if you have three matrices, \( \mathbf{A} \) , \( \mathbf{B} \) , and \( \mathbf{C} \), provided they are of compatible dimensions, the property states that \( (\mathbf{A} \mathbf{B}) \mathbf{C} = \mathbf{A} ( \mathbf{B} \mathbf{C} ) \). This is crucial when solving matrix equations, because it gives us the flexibility to rearrange these equations to achieve a desirable form - one that might be easier to handle or solve. For example, in the exercise, this property allowed for the reassociation of \( \mathbf{B}' \) and \( \mathbf{AB} \) ultimately showing that \( \mathbf{B} = \mathbf{B}' \).

Matrix Equality

Matrix Equality is a term that signifies that two matrices are identical if and only if they have the same dimensions and their corresponding elements are equal. In mathematical terms, two matrices \( \mathbf{A} \) and \( \mathbf{B} \) are equal, \( \mathbf{A} = \mathbf{B} \), if and only if \( a_{ij} = b_{ij} \) for all \( i \) and \( j \). This concept of equality is central to verifying solutions to matrix equations. In the case of the exercise, demonstrating the equality \( \mathbf{B} = \mathbf{B}' \) involves a series of steps that use other properties of matrices to show that every corresponding element in \( \mathbf{B} \) and \( \mathbf{B}' \) must be the same.

Invertible Matrices

An Invertible Matrix, also known as a non-singular or invertible matrix, is one that has an inverse. The inverse of a matrix \( \mathbf{A} \) is another matrix \( \mathbf{A}^{-1} \) such that \( \mathbf{A} \mathbf{A}^{-1} = \mathbf{A}^{-1} \mathbf{A} = \mathbf{I}_n \) where \( \mathbf{I}_n \) is the identity matrix of the same dimension. This concept is important here because the matrices \( \mathbf{B} \) and \( \mathbf{B}' \) both behave as inverses of \( \mathbf{A} \), which leads to the conclusion that they must be equal. Only square matrices can have inverses, and not all square matrices are invertible. The invertibility of a matrix is crucial in many areas, including solving system of linear equations and matrix division.

Identity Matrix Properties

The Identity Matrix, denoted by \( \mathbf{I}_n \), is a special kind of square matrix where all the elements on the main diagonal are 1, and all other elements are 0. It acts as the multiplicative identity in matrix algebra, meaning that for any compatible matrix \( \mathbf{A} \) , multiplying \( \mathbf{A} \) by \( \mathbf{I}_n \) leaves \( \mathbf{A} \) unchanged: \( \mathbf{A} \mathbf{I}_n = \mathbf{I}_n \mathbf{A} = \mathbf{A} \). This property is pivotal in the exercise, as it is used to show that multiplying by the identity does not alter the matrices \( \mathbf{B} \) and \( \mathbf{B}' \) and hence supports the conclusion that these matrices are equal.