Question

Show without computation that the following determinant is equal to zero. Hint: Consider the effect of interchanging rows and columns. $$\left|\begin{array}{rrr} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{array}\right|$$

Short answer

The matrix is skew-symmetric and its order is 3 (an odd number), so its determinant is zero.

Step-by-step solution

- Understand Determinants and Symmetry

Recall that the determinant of a square matrix is zero if the matrix is singular, meaning it lacks linear independence among its rows or columns. One way to show this is by identifying symmetry or patterns in the matrix.

- Investigate the Matrix Structure

Observe the given matrix: \[ \begin{bmatrix} 0 & 2 & -3 \ -2 & 0 & 4 \ 3 & -4 & 0 \end{bmatrix} \]Notice that by interchanging rows and columns, you get a similar matrix structure, particularly focusing on the signs and symmetry patterns.

- Recognize the Skew-Symmetric Property

Identify the given matrix as a skew-symmetric matrix because it satisfies the condition \( A^T = -A \). For skew-symmetric matrices of odd order (3x3 in this case), their determinant is always zero.

- Conclude the Determinant Value

Since the given matrix is a 3x3 skew-symmetric matrix, it inherently has a determinant of zero due to its odd order.

Key concepts

skew-symmetric matrix

In the study of matrices, a skew-symmetric matrix is an important concept. It is defined by the property that its transpose is equal to its negative. Mathematically, if a matrix \(A\) is skew-symmetric, then it satisfies:
\( A^T = -A \).
For example, consider the matrix given in the exercise:
\[ \begin{bmatrix} 0 & 2 & -3 \ -2 & 0 & 4 \ 3 & -4 & 0 \end{bmatrix} \]
Notice that by taking the transpose \(A^T\) (interchanging rows and columns), and comparing it to \(-A\) (negating each element), we have:
\[ \begin{bmatrix} 0 & -2 & 3 \ 2 & 0 & -4 \ -3 & 4 & 0 \end{bmatrix} \equiv \begin{bmatrix} 0 & 2 & -3 \ -2 & 0 & 4 \ 3 & -4 & 0 \end{bmatrix} \]
This confirms that our matrix is indeed skew-symmetric. An interesting property of skew-symmetric matrices is that if the order (size) of the matrix is odd, its determinant will always be zero. This proof does not require computation but relies purely on the structural properties of the matrix.

linear independence

Linear independence is another key concept when working with determinants and matrices. It refers to the condition where no row or column in the matrix can be written as a linear combination of the other rows or columns. If any row or column can be expressed as such a combination, the determinant of the matrix is zero.
In our original problem, the determinant of the matrix was found to be zero without computation due to its skew-symmetric property. However, this zero determinant also signifies that there is a linear dependence among the rows and columns of the matrix. This means that the vectors (rows or columns) are not linearly independent. This lack of linear independence helps reinforce our previous conclusion about the determinant being zero.
In other words, if the rows or columns of a matrix cannot span the entire space (in this case, \(\textbf{R}^3\)), the matrix is singular, leading to a zero determinant.

matrix symmetry

Matrix symmetry plays a crucial role in simplifying problems involving determinants. While skew-symmetric matrices (like the one given in the exercise) are a specific type of symmetry, other forms of symmetry can also be incredibly useful.
A matrix is symmetric if it equals its transpose: \(A = A^T\). This is different from skew-symmetric matrices where \(A^T = -A\). Symmetric matrices have many useful properties, such as real eigenvalues and orthogonal eigenvectors.
Recognizing symmetry or patterns in a matrix can often reveal properties that are not immediately obvious. For instance, it can lead to simplified computations of determinants, ease the solving of linear systems, and help identify invariants under certain transformations.
In the context of our exercise, recognizing the skew-symmetric nature saved us from lengthy calculations by directly indicating that the determinant must be zero due to the matrix's odd order. Understanding these symmetries can thus provide significant insights and shortcuts in various linear algebra problems.