Question

In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Multiplication of a nonsingular matrix and its inverse is commutative.

Short answer

The statement is true. The multiplication of a nonsingular matrix and its inverse is commutative.

Step-by-step solution

Understanding the term 'Nonsingular Matrix'

A nonsingular matrix or invertible matrix or nondegenerate matrix is a square matrix that does not have a matrix inverse. An n-by-n square matrix A is said to be invertible if there exists another n-by-n matrix such that \(AA^{-1}=A^{-1}A=I_n\), where \(I_n\) denotes the n-by-n identity matrix.

Check the Commutative property for Matrix multiplication

Matrix multiplication doesn’t generally obey the commutative law. In other words, for two matrices A and B, it is not always true that AB = BA. However, for any nonsingular matrix A and its inverse \(A^{-1}\), it is always true that \(AA^{-1}=A^{-1}A=I_n\). This is a special case because we are multiplying a matrix with its inverse, which always results in the identity matrix, regardless of the order of multiplication.

Conclusion

So, the statement 'Multiplication of a nonsingular matrix and its inverse is commutative' is true. The commutative property holds for a matrix and its inverse.

Key concepts

Nonsingular Matrix

A nonsingular matrix, also known as an invertible or nondegenerate matrix, plays a crucial role in linear algebra. It is a square matrix that has an inverse, meaning another matrix exists that can "undo" the operations of the original matrix.
A square matrix is defined as nonsingular if it satisfies the following condition: for an n-by-n matrix \( A \), there exists an n-by-n matrix \( A^{-1} \) such that when they are multiplied together in either order, they produce the identity matrix \( I_n \). The identity matrix is a special kind of square matrix characterized by ones on the diagonal from the top left to the bottom right and zeros elsewhere.
This property makes nonsingular matrices quite significant because their existence implies that matrix equations of the form \( AX = B \) can be solved uniquely, where \( X \) is the unknown matrix to be determined. Understanding the nature of nonsingular matrices is fundamental when exploring systems of linear equations.

Inverse Matrix

The concept of an inverse matrix is essential for understanding solutions to matrix equations. If a matrix \( A \) is invertible, there exists another matrix, denoted as \( A^{-1} \), such that their product, in either order, results in the identity matrix: \( AA^{-1} = A^{-1}A = I_n \).
This is similar to the reciprocal function for real numbers, where multiplying a number by its reciprocal results in one. An inverse matrix effectively "undoes" the transformation imposed by its original matrix. Not all matrices are invertible.
To determine if a matrix has an inverse, it's important to check its determinant. For a matrix to be invertible, the determinant must not be zero. If the determinant is zero, the matrix is said to be singular, meaning it does not have an inverse and cannot be used to "reverse" matrix operations in the context of solving matrix equations.

Commutative Property

The commutative property is a basic principle in algebra, suggesting that the order in which two numbers or variables are multiplied does not affect the product. While this property holds true for arithmetic operations like addition and multiplication of numbers, it fails to apply universally in matrix multiplication.
Generally, for matrices \( A \) and \( B \), the products \( AB \) and \( BA \) are not equal. However, an interesting exception arises when considering a nonsingular matrix and its inverse. In this special case, the commutative property does hold: \( AA^{-1} = A^{-1}A = I_n \). This happens because the product results in the identity matrix, regardless of multiplication order.
This unique property emphasizes how special the relationship is between a nonsingular matrix and its inverse, making the understanding of matrix inverses crucial for various applications in linear algebra. However, it is important to remember that this is a specific exception and not the general rule for matrix multiplication.