Problem 22 - Advanced Engineering Mathematics

Question

Suppose \(\mathbf{A}\) is an \(n \times n\) matrix such that \(\mathbf{A}^{2}=\mathbf{I}\) where \(\mathbf{A}^{2}=\mathbf{A} \mathbf{A} .\) Show that \(\operatorname{det} \mathbf{A}=\pm 1\).

Step-by-Step Solution

Answer

The determinant of \( \mathbf{A} \) is \( \pm 1 \).

Step 1: Understand the Given Statement

We are given that \( \mathbf{A} \) is an \( n \times n \) matrix and that \( \mathbf{A}^2 = \mathbf{I} \), which means that multiplying \( \mathbf{A} \) by itself results in the identity matrix \( \mathbf{I} \). Our task is to find the possible values for the determinant of \( \mathbf{A} \).

Step 2: Consider Properties of Determinants

We know that for any two \( n \times n \) matrices \( \mathbf{B} \) and \( \mathbf{C} \), the determinant of their product is the product of their determinants. That is, \( \operatorname{det}(\mathbf{B}\mathbf{C}) = \operatorname{det}(\mathbf{B})\operatorname{det}(\mathbf{C}) \).

Step 3: Apply Determinant to Both Sides of \( \mathbf{A}^2 = \mathbf{I} \)

Applying the determinant to both sides of the equation \( \mathbf{A}^2 = \mathbf{I} \) gives us:\[\operatorname{det}(\mathbf{A}^2) = \operatorname{det}(\mathbf{I})\]Since \( \operatorname{det}(\mathbf{I}) = 1 \) for any identity matrix, we have:\[\operatorname{det}(\mathbf{A}^2) = 1\]

Step 4: Use the Property of Determinants

From Step 3, \( \operatorname{det}(\mathbf{A}^2) = (\operatorname{det}(\mathbf{A}))^2 = 1 \). Since the square of the determinant is 1, solving \( (\operatorname{det}(\mathbf{A}))^2 = 1 \) gives two possible solutions: \( \operatorname{det}(\mathbf{A}) = 1 \) or \( \operatorname{det}(\mathbf{A}) = -1 \).

Step 5: Conclusion

The possible values for \( \operatorname{det}(\mathbf{A}) \) are \( 1 \) or \( -1 \). Therefore, we conclude \( \operatorname{det}(\mathbf{A}) = \pm 1 \).

Key Concepts

Matrix Algebra

Matrix algebra is a system of mathematics that extends many of the concepts of arithmetic to arrays of numbers called matrices. This algebra provides powerful tools to solve systems of linear equations, perform transformations, and work with vector spaces. In the context of the given problem, we are dealing with an interesting property of matrices where a matrix squared equals the identity matrix. This scenario commonly arises in problems involving symmetry and transformations.

When we multiply two matrices, it's important to remember that the operation is not commutative. This means that multiplying matrix A with matrix B can yield a different result than multiplying matrix B with matrix A. However, the determinant, which is a special number associated with a square matrix, follows the multiplicative property. This property states:

• If \( \mathbf{B} \) and \( \mathbf{C} \) are two \( n \times n \) matrices, then \( \operatorname{det}(\mathbf{B}\mathbf{C}) = \operatorname{det}(\mathbf{B})\operatorname{det}(\mathbf{C}) \).

This is crucial when dealing with our specific exercise, as it allows us to deduce that since \( \mathbf{A}^2 = \mathbf{I} \), we can say \( \operatorname{det}(\mathbf{A}^2) = \operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{A}) = (\operatorname{det}(\mathbf{A}))^2 \). By calculating the determinant for both sides of this equation, we can gain insight into the determinants' values.

Identity Matrix

An identity matrix, often symbolized by \( \mathbf{I} \), holds a significant place in matrix algebra. It acts like the number 1 in basic arithmetic, serving as a neutral element. For any square matrix \( \mathbf{A} \), when you multiply \( \mathbf{A} \) with an identity matrix of the same dimension, the result is \( \mathbf{A} \) itself:

• \( \mathbf{A} \cdot \mathbf{I} = \mathbf{I} \cdot \mathbf{A} = \mathbf{A} \)

This property makes the identity matrix similar to the number one, hence its significance in verifying solutions and transformations within matrix equations.

In our exercise, the equation \( \mathbf{A}^2 = \mathbf{I} \) reveals that multiplying the matrix \( \mathbf{A} \) by itself yields the identity matrix, suggesting that \( \mathbf{A} \) has an inverse which is itself. This property simplifies the process of calculating the determinant because we already know that the determinant of the identity matrix \( \operatorname{det}(\mathbf{I}) \) is always equal to 1, regardless of its size. Knowing this fact allows us to conclude that the determinant of the squared matrix \( \mathbf{A}^2 \), and thus \( (\operatorname{det}(\mathbf{A}))^2 \), must also be equal to 1.

Matrix Properties

Matrices possess unique properties that make them useful and versatile in solving numerous mathematical problems. Some key properties are:

• Determinants: Indicating the scaling factor of the matrix transformation.
• Inverses: A matrix that results in an identity matrix when multiplied by the original matrix.
• Transpose: A significant transformation that flips a matrix over its diagonal.

For our specific task, we exploit a crucial property, which is the determinant of a matrix product. Because we have \( \mathbf{A}^2 = \operatorname{I} \), and we apply the determinant rule \( \operatorname{det}(\mathbf{A})^2 = \operatorname{det}(\mathbf{I}) \). This points us directly to finding the possible determinant values for \( \mathbf{A} \).

Another interesting aspect is the notion that any matrix that satisfies \( \mathbf{A}\cdot \mathbf{A} = \mathbf{I} \) is called an involutory matrix. These matrices have the characteristic that their determinant can only be \(+1\) or \(-1\), which links directly back to our initial solution. Thus, by understanding these basic matrix properties, we can leverage them to solve more complex problems like the one presented in the exercise.