Problem 21 - Differential Equations and Linear Algebra

Question

If \(A\) is an invertible \(n \times n\) matrix, prove property \(\mathrm{P} 9:\) $$\operatorname{det}\left(A^{-1}\right)=\frac{1}{\operatorname{det}(A)}.$$

Step-by-Step Solution

Answer

To prove that \(\operatorname{det}\left(A^{-1}\right)=\frac{1}{\operatorname{det}(A)}\), we start with the equation \(A A^{-1} = I\). Then, we multiply both sides of the equation by their determinants and apply the property \(\text{det}(AB) = \text{det}(A) \text{det}(B)\): \[\text{det}(A) \text{det}(A^{-1}) = \text{det}(I)\] Since \(\text{det}(I) = 1\), we obtain: \[\text{det}(A) \text{det}(A^{-1}) = 1\] Solving for \(\text{det}(A^{-1})\) gives us the desired result: \[\text{det}\left(A^{-1}\right) = \frac{1}{\text{det}(A)}\]

Step 1: Recall the definition of the inverse matrix

The inverse of a matrix A, denoted as \(A^{-1}\), is the unique matrix that, when multiplied by A, results in the identity matrix I, i.e., \(A A^{-1} = A^{-1} A = I\). If such a matrix exists, then A is called invertible.

Step 2: Recall the properties of determinants

For any two n × n matrices A and B, we have the following properties of determinants: 1. \(\text{det}(AB) = \text{det}(A) \text{ det}(B)\) 2. \(\text{det}(I) = 1\)

Step 3: Apply the determinant properties to the inverse matrix

To establish the given property, let's multiply both sides of the equation \(A A^{-1} = I\) by their determinants and apply property 1 of determinants: \[\text{det}(A A^{-1}) = \text{det}(I)\] According to property 2, the determinant of the identity matrix is 1. So, we have: \[\text{det}(A) \text{det}(A^{-1}) = 1\]

Step 4: Solve for the determinant of the inverse matrix

To find an expression for \(\text{det}(A^{-1})\), we need to solve the equation obtained in step 3 for \(\text{det}(A^{-1})\): \[\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}\] This is the expression for the determinant of the inverse matrix, and it proves the given property: \[\text{det}\left(A^{-1}\right) = \frac{1}{\text{det}(A)}\]

Key Concepts

Determinant Properties

Determinant properties play a crucial role in understanding inverses of matrices. One of the fundamental properties is how the determinant behaves under multiplication. Given two matrices, say \( A \) and \( B \), the determinant of their product (\( AB \)) is simply the product of their determinants: \( \text{det}(AB) = \text{det}(A) \times \text{det}(B) \). This highlights how determinants interact with multiplication, making them a powerful tool in linear algebra.

Another vital property is the determinant of the identity matrix, denoted as \( I \). The determinant of \( I \) is always 1, irrespective of the size of the identity matrix. This constant value is tremendously useful, especially when working with inverse matrices, as it helps in establishing other properties, such as the one we aim to prove: \( \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \). To arrive at this outcome, we use these pervading properties that simplify complex expressions into manageable computations.

Matrix Multiplication

Matrix multiplication is a cornerstone operation in linear algebra. It involves taking the dot product of rows and columns from two matrices to produce another matrix. When dealing with square matrices of the same size, such as \( A \) and \( A^{-1} \), the operation generates the identity matrix. Specifically, \( AA^{-1} = I \) and \( A^{-1}A = I \). This equation is pivotal in proving properties about determinants and inverse matrices.

It’s important to understand that matrix multiplication is not commutative. This means that \( AB \) does not necessarily equal \( BA \). However, associativity still holds, meaning the product order can be rearranged without changing the result: \((AB)C = A(BC)\). Recognizing these properties aids greatly in performing and mastering matrix operations in various mathematical and computational problems.

Identity Matrix

An identity matrix is a special kind of square matrix that acts like the number 1 in matrix algebra. For any matrix \( A \), the identity matrix, denoted by \( I \), satisfies \( AI = A \) and \( IA = A \). It's like a neutral element of multiplication, which makes it essential in defining inverse matrices. When \( A \) is multiplied by its inverse \( A^{-1} \), they yield the identity matrix: \( AA^{-1} = I \) and \( A^{-1}A = I \).

This property is instrumental in proving the relation between the determinants of \( A \) and \( A^{-1} \). With the determinant of the identity matrix being 1, it swiftly links the concept of inverse matrices and helps establish that the determinant of an inverse is indeed \( \frac{1}{\text{det}(A)} \). The identity matrix, with its straightforward properties, effectively bridges various aspects of matrix theory, making operations like solving linear equations much simpler.