Problem 35 - Advanced Engineering Mathematics

Question

If \(\mathbf{A}\) and \(\mathbf{B}\) are nonsingular \(n \times n\) matrices, use Theorem 8.6.3 to show that \(\mathbf{A B}\) is nonsingular.

Step-by-Step Solution

Answer

By Theorem 8.6.3, if \( \mathbf{A} \) and \( \mathbf{B} \) are nonsingular, then \( \mathbf{AB} \) is nonsingular, with inverse \( (\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1} \).

Step 1: Understand the Concept

To show that the matrix \( \mathbf{AB} \) is nonsingular, we need to demonstrate that \( \mathbf{AB} \) has an inverse. According to Theorem 8.6.3, if two matrices \( \mathbf{A} \) and \( \mathbf{B} \) are nonsingular, then their product should also be nonsingular.

Step 2: Recall Definition of Nonsingular Matrices

A matrix is nonsingular if it has an inverse. For \( \mathbf{A} \) and \( \mathbf{B} \), this means there are matrices \( \mathbf{A}^{-1} \) and \( \mathbf{B}^{-1} \) such that \( \mathbf{A} \mathbf{A}^{-1} = \mathbf{I} \) and \( \mathbf{B} \mathbf{B}^{-1} = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix.

Step 3: Use Theorem 8.6.3

Theorem 8.6.3 states that the product of two nonsingular matrices is also nonsingular. This theorem also gives the inverse of the product as: \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\).

Step 4: Verify Inverse Exists for Product

To verify, compute \( \mathbf{AB}(\mathbf{B}^{-1}\mathbf{A}^{-1}) = \mathbf{A}(\mathbf{BB}^{-1})\mathbf{A}^{-1} = \mathbf{A}\mathbf{I}\mathbf{A}^{-1} = \mathbf{AA}^{-1} = \mathbf{I} \). Similarly, check \( (\mathbf{B}^{-1}\mathbf{A}^{-1})\mathbf{AB} = (\mathbf{B}^{-1}(\mathbf{A}^{-1}\mathbf{A}))\mathbf{B} = (\mathbf{B}^{-1}\mathbf{I})\mathbf{B} = \mathbf{I}\). As both are true, \( \mathbf{AB} \) has an inverse.

Key Concepts

Inverse Matrices

An inverse matrix is an essential concept in linear algebra, especially when dealing with nonsingular matrices. A matrix is said to be nonsingular if it possesses an inverse. This means there exists another matrix of the same dimensions, called the inverse matrix, that, when multiplied by the original matrix, results in the identity matrix. For a given matrix \( \mathbf{A} \), its inverse is denoted as \( \mathbf{A}^{-1} \), satisfying the equation \( \mathbf{A} \mathbf{A}^{-1} = \mathbf{A}^{-1} \mathbf{A} = \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix.
Determining if a matrix has an inverse involves checking whether the determinant of the matrix is non-zero. A zero determinant indicates the matrix is singular, meaning it does not have an inverse. The concept of inverse matrices is crucial for solving systems of linear equations and in various applications in physics and engineering.
To better understand, remember these characteristics of inverse matrices:

• The product of a matrix and its inverse is always the identity matrix.
• Not all matrices have inverses; only nonsingular (invertible) matrices have inverses.
• The inverse of a matrix is unique.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that represents the product of two matrices. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. This operation combines the rows of the first matrix with the columns of the second matrix to produce a new matrix.
The product of two matrices \( \mathbf{A} \) and \( \mathbf{B} \), represented as \( \mathbf{AB} \), is calculated by taking the dot product of each row of \( \mathbf{A} \) with each column of \( \mathbf{B} \). The resulting element in the product matrix is the sum of these products.
It is important to note specific properties of matrix multiplication:

• Matrix multiplication is not commutative: \( \mathbf{AB} eq \mathbf{BA} \) in general.
• It is associative: \( (\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C}) \).
• It distributes over addition: \( \mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC} \).

This operation is widely used in areas such as computer graphics and transformations, where combined transformations are represented by the multiplication of matrices.

Identity Matrix

The identity matrix is a special kind of square matrix that acts as the multiplicative identity in the matrix world. Just like multiplying any number by one leaves the number unchanged, multiplying any matrix by the identity matrix returns the original matrix. This unique property is why it is called the 'identity matrix'.
An identity matrix of size \( n \times n \) is denoted by \( \mathbf{I}_n \). It is composed of ones on its main diagonal (from top left to bottom right) and zeros elsewhere. For example, a 3x3 identity matrix looks like this: \[ \mathbf{I}_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{pmatrix} \]The identity matrix retains the original matrix when used in multiplication:

• On the right: \( \mathbf{A} \mathbf{I} = \mathbf{A} \)
• On the left: \( \mathbf{I} \mathbf{A} = \mathbf{A} \)

Identity matrices are significant in defining inverses, as the product of a matrix and its inverse is always the identity matrix. They are instrumental in simplifying matrix expressions and solving algebraic equations involving matrices.