Problem 37 - Advanced Engineering Mathematics

Question

Show that if \(\mathbf{A}\) is a nonsingular matrix, then \(\operatorname{det} \mathbf{A}^{-1}=1 /\) det \(\mathbf{A}\).

Step-by-Step Solution

Answer

\(\operatorname{det}(\mathbf{A}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{A})}\).

Step 1: Understand matrix properties

Recall that a matrix \(\mathbf{A}\) is nonsingular (invertible) if and only if its determinant is non-zero. The inverse of a matrix \(\mathbf{A}\), denoted by \(\mathbf{A}^{-1}\), satisfies \(\mathbf{A} \mathbf{A}^{-1} = \mathbf{I}\), where \(\mathbf{I}\) is the identity matrix.

Step 2: Use determinant property of products

One important property of determinants is that for any two square matrices \(\mathbf{A}\) and \(\mathbf{B}\), \(\operatorname{det}(\mathbf{A} \mathbf{B}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{B})\). In our context, this means \(\operatorname{det}(\mathbf{A} \mathbf{A}^{-1}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{A}^{-1})\).

Step 3: Calculate determinant of identity matrix

Since \(\mathbf{A} \mathbf{A}^{-1} = \mathbf{I}\), apply the property from the previous step: \(\operatorname{det}(\mathbf{A} \mathbf{A}^{-1}) = \operatorname{det}(\mathbf{I})\). The determinant of the identity matrix is always 1, so we have \(\operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{A}^{-1}) = 1\).

Step 4: Solve for \(\operatorname{det}(\mathbf{A}^{-1})\)

From the equation \(\operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{A}^{-1}) = 1\), solve for \(\operatorname{det}(\mathbf{A}^{-1})\) by dividing both sides by \(\operatorname{det}(\mathbf{A})\). This yields \(\operatorname{det}(\mathbf{A}^{-1}) = \frac{1}{\operatorname{det}(\mathbf{A})}\).

Key Concepts

Determinant Properties

Determinants have fascinating properties that are central to understanding linear transformations in linear algebra. They are numerical values calculated from a square matrix and provide key insights into matrix characteristics.

A few cornerstone properties include:

• The determinant of a product of matrices equals the product of their determinants: \( \operatorname{det}(\mathbf{AB}) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{B}) \).
• If a matrix is invertible (or nonsingular), its determinant is non-zero.
• The determinant of the identity matrix is always 1.
• A matrix is not invertible (singular) if its determinant is zero.

By using these properties, we gain the ability to maneuver through matrix equations, predict behaviors, and solve complex linear transformations with greater ease.

Nonsingular Matrix

A nonsingular matrix, often called an invertible matrix, is a key player in linear algebra. These matrices are characterized by a few important features.

• Their determinant is non-zero, which implies that they have an inverse.
• They represent linear transformations that are bijective, meaning they have a one-to-one mapping.

The existence of an inverse is crucial for solving systems of linear equations, as it guarantees a unique solution. It's important to note that only square matrices can be nonsingular, as only they can have inverses. The inverse matrix, denoted by \( \mathbf{A}^{-1} \), satisfies \( \mathbf{AA}^{-1} = \mathbf{I} \). Understanding nonsingularity helps us confirm matrix invertibility and is foundational for many higher-level concepts in linear algebra.

Identity Matrix

The identity matrix, typically denoted as \( \mathbf{I} \), is a special matrix. In many ways, it acts like the number 1 in matrix multiplication.

• For any square matrix \( \mathbf{A} \), multiplying by the identity matrix leaves it unchanged: \( \mathbf{AI} = \mathbf{IA} = \mathbf{A} \).
• It has 1's on its main diagonal and 0's elsewhere.
• Its determinant is always 1, regardless of its size.

The identity matrix is fundamental in defining the inverse of a matrix. If a matrix \( \mathbf{A} \) has an inverse \( \mathbf{A}^{-1} \), then multiplying \( \mathbf{A} \) by \( \mathbf{A}^{-1} \) results in the identity matrix, \( \mathbf{AA}^{-1} = \mathbf{I} \). This property facilitates the simplification of many matrix equations and plays a crucial role in inverse calculation.

Linear Algebra Concepts

Linear algebra is a branch of mathematics that studies vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.

Here are some pivotal concepts:

• **Vectors and Vector Spaces:** The fundamental building blocks of linear algebra, vectors can be thought of as points or arrows in space, and vector spaces are collections of vectors that can be added together and multiplied by scalars.
• **Linear Transformations:** Functions between vector spaces that preserve vector addition and scalar multiplication. They can be represented as matrices.
• **Systems of Linear Equations:** These can be solved using methods such as Gaussian elimination and can be represented in matrix form.

Mastery of linear algebra concepts enables us to solve problems related to computer graphics, engineering, machine learning, and more. It forms the mathematical backbone of these and many other fields.

Square Matrices

Square matrices are matrices with the same number of rows and columns, denoted as \( n \times n \). Their symmetrical nature allows them to have unique properties, especially in matrix algebra.

• They can have determinants, which help determine characteristics of the matrix such as invertibility.
• They can be nonsingular or singular, with nonsingular matrices having inverses.
• Many types of products, such as eigenvectors and eigenvalues, are strictly defined for square matrices.
• The identity matrix itself is always a square matrix.

Square matrices are indispensable in studying transformations, solving linear equations, and exploring eigenvectors. Their balanced structure is crucial for various algebraic operations, making them a foundational concept in linear algebra.