Question

Explain what can be said about \(\operatorname{det} A\) if: a. \(A^{2}=A\) b. \(A^{2}=I\) c. \(A^{3}=A\) d. \(P A=P\) and \(P\) is invertible e. \(A^{2}=u A\) and \(A\) is \(n \times n\) f. \(A=-A^{T}\) and \(A\) is \(n \times\) \(n\) g. \(A^{2}+I=0\) and \(A\) is \(n \times n\)

Short answer

a. 0 or 1, b. ±1, c. 0, d. 1, e. 0, f. 0 (n odd), g. Non-zero (n even).

Step-by-step solution

- Analyze property a: Idempotent matrix

Given that \(A^2 = A\), this implies that \(A(A - I) = 0\). Therefore, \(A\) is an idempotent matrix. The determinant of an idempotent matrix is either 0 or 1. Thus, \(\operatorname{det} A = 0\) or \(1\).

- Analyze property b: Involutory matrix

Given that \(A^2 = I\), this implies that \(A\) is an inverse of itself. The determinant of the identity matrix \(I\) is 1, so \(\operatorname{det} A = \pm 1\) because \(\det(A^2) = (\operatorname{det} A)^2 = 1\).

- Analyze property c: Cubic identity

Given that \(A^3 = A\), factor the equation to \(A(A^2 - I) = 0\), meaning any of \(A\) or \(A^2 - I = 0\) must have eigenvalue zero. Hence, \(\operatorname{det} A = 0\) is possible or not necessarily zero, but further details about \(A\) are needed for a precise answer.

- Analyze property d: Left-multiplication identity

Given that \(PA = P\) and \(P\) is invertible, we can multiply both sides by \(P^{-1}\) to obtain \(A = I\). Thus, \(\operatorname{det} A = 1\) for the identity matrix.

- Analyze property e: Scalar matrix

Given \(A^2 = uA\), we can rearrange to \(A(A - uI) = 0\). This implies \(A - uI\) and \(A\) might have zero determinants. Thus, \(\operatorname{det} A = 0\) or \(\operatorname{det} (A-uI) = 0\) is at play, further implying \(\operatorname{det} A = 0\).

- Analyze property f: Skew-symmetric matrix

Given \(A = -A^T\), skew-symmetric matrices have determinants that are zero when \(n\) is odd since the eigenvalues are purely imaginary and appear in conjugate pairs. Thus, \(\operatorname{det} A = 0\) when \(n\) is odd; \(\operatorname{det} A \geq 0\) when \(n\) is even.

- Analyze property g: Negative identity

Given that \(A^2 + I = 0\), this implies \(A^2 = -I\). The determinant of \(-I\) is \((-1)^n\), so \(\operatorname{det} A = \pm i^n\) which is real and non-zero only if \(n\) is even. Hence, \(\operatorname{det} A eq 0\).

Key concepts

Idempotent Matrix

An idempotent matrix is an intriguing type of matrix that remains unchanged when multiplied by itself. Mathematically, for a matrix \( A \), if \( A^2 = A \), then \( A \) is idempotent. This property has a significant impact on the matrix's determinant. The determinant of an idempotent matrix is always 0 or 1. Why? Since \( A(A - I) = 0 \) must hold, it implies that either \( A \) itself is zero or contains at least one eigenvalue that is 0 (leading to \( \operatorname{det}(A) = 0 \)), or \( A \) is the identity matrix (thus \( \operatorname{det}(A) = 1 \)). These matrices are particularly useful in statistics, especially in regression analysis, where projections onto regression manifolds yield idempotent matrices.

Involutory Matrix

An involutory matrix is a fascinating concept in linear algebra. A matrix \( A \) is termed involutory if when squared, it returns the identity matrix, i.e., \( A^2 = I \). Such matrices essentially act as their own inverses. With respect to determinants, because the determinant of an identity matrix \( I \) is 1, for any involutory matrix \( \operatorname{det}(A) = \pm 1 \). This stems from the property \( \operatorname{det}(A^2) = (\operatorname{det}(A))^2 = 1 \), so \( \operatorname{det}(A) \) must be \( \pm 1 \). Involutory matrices are interesting not only in mathematical theory but also in practical applications such as image processing, where specific involutory transformations can be used to manipulate image symmetries.

Skew-Symmetric Matrix

Skew-symmetric matrices offer unique properties in linear algebra. A matrix \( A \) is skew-symmetric if \( A = -A^T \). This implies that \( A \) is equal to the negative of its transpose. One core property of skew-symmetric matrices is their determinant, which is always zero if the order \( n \) of \( A \) is odd. This arises because the eigenvalues of skew-symmetric matrices are purely imaginary numbers that, when paired, need to multiply to zero. For even \( n \), the determinant can still be zero or non-negative. These matrices have applications across various fields, including control theory and physics, particularly in describing rotational behaviors and angular momenta.

Eigenvalues

Eigenvalues are fundamental to understanding the behavior of matrices. For any square matrix \( A \), the eigenvalues are found by solving the characteristic equation \( \operatorname{det}(A - \lambda I) = 0 \), where \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix. These values reveal much about the matrix's properties. For instance, the determinant of a matrix is the product of its eigenvalues. If one of the eigenvalues is zero, then the determinant is zero as well. Eigenvalues have deep implications in systems stability, vibrations in mechanical systems, and stability in differential equations, indicative of how a system behaves over time.

Identity Matrix

The identity matrix is a pivotal concept in linear algebra. It is a square matrix with ones on the diagonal and zeroes elsewhere. Denoted by \( I \), an identity matrix of size \( n \times n \) is such that for any compatible matrix \( A \), the product \( AI = IA = A \). Its determinant is always 1, reflecting its role as a multiplicative identity in matrix algebra. The identity matrix is crucial because it serves as the equivalent of 1 in matrix operations, anchoring the concept of matrix inversion and transformations. Identity matrices are also foundational in defining and identifying other matrix types, such as involutory matrices, where \( A^2 = I \). They are ubiquitous in mathematical computations and applications that require preserving the properties of objects under transformations.