(a) Show that if A has size n ×n, then det(-A)=(-1)^n det A. (b) Prove that, for n odd, any n ×n

step 1 Section 3 .2, problem number 59, we're given that A is an N -by -N -N -skew symmetric matrix, an

(a) Show that if A has size n ×n, then det(-A)=(-1)^n det A....

(a) Show that if $A$ has size $n \times n$, then $\operatorname{det}(-A)=(-1)^n \operatorname{det} A$. (b) Prove that, for $n$ odd, any $n \times n$ skew-symmetric matrix $A=-A^T$ is singular. (c) Find a nonsingular skew-symmetric matrix.

Key Concepts

Skew-Symmetric Matrices

A skew-symmetric matrix is defined by the property that its transpose equals the negative of the matrix, i.e., A^T = -A. This symmetry property has profound implications for the determinants and eigenvalues of the matrix. In particular, it leads to interesting results regarding the singularity and non-singularity of such matrices in relation to their size (order).

Determinant of Scalar Multiples

This concept involves understanding how the determinant of a matrix behaves when the matrix is multiplied by a scalar. Specifically, if each entry of an n×n matrix is multiplied by a scalar t, the determinant of the resulting matrix is t^n times the determinant of the original matrix. This property is pivotal for analyzing expressions like det(?A), as it immediately leads to det(?A) = (?1)^n det A.

Matrix Singularity

A matrix is called singular if it does not have an inverse, which is equivalent to having a determinant equal to zero. The investigation of singularity is central to linear algebra, as it determines whether a matrix equation has a unique solution. In the context of skew-symmetric matrices, the property det(?A) = (?1)^n det A creates a condition that forces the determinant to be zero when n is odd, rendering the matrix singular.

Parity of Matrix Order

The parity (odd or even nature) of the order of a matrix plays a critical role in the properties of determinants and the structure of matrices such as skew-symmetric matrices. For instance, when n is odd, the equation det(A) = ?det(A) that arises from the skew-symmetric property necessitates that det(A) must be zero, whereas when n is even, nonzero determinants (and thus nonsingular matrices) are possible. This distinction informs many theoretical and practical applications in linear algebra.

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