Question

If \(A\) and \(B\) are \(n \times n\) matrices, if \(A B=\) \(-B A,\) and if \(n\) is odd, show that either \(A\) or \(B\) has no inverse.

Short answer

Either \(A\) or \(B\) must be non-invertible if \(n\) is odd.

Step-by-step solution

Understand the Properties

Two matrices, \(A\) and \(B\), are given such that \(AB = -BA\). We need to show that if \(n\), the size of the square matrices, is odd, then at least one of the matrices, \(A\) or \(B\), is not invertible.

Assume Contradiction

Assume for the sake of contradiction that both \(A\) and \(B\) are invertible. Our goal is to derive a contradiction from this assumption given the properties of the matrices.

Multiply by Inverses

Because \(A\) and \(B\) are assumed to be invertible, consider the product of the matrices and their inverses: \( (A^{-1}B^{-1})(AB) = A^{-1}(BA)B^{-1} \). Simplifying this as \((A^{-1} B^{-1}) (AB) = -1\), as substituting from the property \(AB = -BA\), we find \( (I)(I) = -I \).

Analyze the Determinant

Since \( (A^{-1} B^{-1}) (AB) = -I \) implies \( I = -I \), taking determinants on both sides gives \( \det(-I) = (-1)^n \det(I) \). But \( \det(I) = 1 \) and \( \det(-I) = (-1)^n \).

Use Odd Property

Recall that \(n\) is odd, so \((-1)^n = -1\), meaning \(\det(-I) = -1\). Similarly \(\det(I) = 1\). Thus, equating \(\det(I)\) and \(\det(-I)\) yields \(1 = -1\), a contradiction, which is impossible. Therefore, at least one matrix is not invertible.

Key concepts

Determinants

Determinants are a crucial concept in matrix algebra, primarily used to determine whether a matrix is invertible. For a square matrix, the determinant is a scalar value that can be calculated using a specific formula depending on the matrix's size. If the determinant of a matrix is zero, the matrix is non-invertible (singular); otherwise, it's invertible (non-singular). Let's explore why determinants are essential:

• Non-zero Determinant: If the determinant of a matrix is non-zero, the matrix has an inverse. This property is particularly useful for solving systems of linear equations using matrix methods.
• Zero Determinant: A zero determinant means the matrix cannot be inverted, indicating that the rows or columns are linearly dependent (not full rank).
• Interaction with Matrix Product: For two matrices, \(A\) and \(B\), the determinant of their product is the product of their determinants (\(\det(AB) = \det(A)\det(B)\)).

These properties help understand the step-by-step solution of the exercise. We used the property \(\det(AB) = \det(-BA)\) for odd \(n\) to prove a contradiction. With odd dimensions, certain determinant properties become pivotal in proving the non-invertibility of at least one matrix.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and operations on them. Understanding the rules of matrix multiplication, inversion, and determinants is key to solving complex problems involving matrices. Here are some foundational concepts of matrix algebra to remember:

• Matrix Multiplication: The product of two matrices depends on the arrangement and dimensions. For \(AB = -BA\), the commutative property does not hold, which is significant in our solution.
• Inverse of a Matrix: The inverse of a matrix \(A\) is another matrix \(A^{-1}\) such that \(AA^{-1} = I\), where \(I\) is the identity matrix. Not all matrices, particularly those with zero determinants, have inverses.
• Algebraic Properties: Understanding properties like associativity and distributivity in matrices aids in simplifying and solving equations, as seen when manipulating \((A^{-1}B^{-1})(AB) = -1\).

Matrix algebra helps in understanding the transformations and solutions of linear equations using matrices, effectively applying it to show non-invertibility in the given problem.

Odd Dimensions

Odd dimensions, or matrices with an odd size \(n\), play a unique role when dealing with determinants and matrix properties. Matrices distinguished by odd or even dimensions can have different properties:

• Determinant Signs: For an identity matrix \(I_n\), if \(n\) is odd, \((-1)^n = -1\), which gives \(\det(-I) = -1\). This is key in deriving contradictions for invertibility as shown in our problem.
• Matrix Behavior: With odd dimensions, certain algebraic conclusions can be drawn about matrix transformations and properties, which differ from even dimensions. This was leveraged to prove that at least one of \(A\) or \(B\) must be non-invertible when \(n\) is odd.

Understanding the effects of matrix dimensions helps in analyzing specific properties and behaviors, such as how determinant signs vary, which is crucial to our matrix problem-solving approaches.