Question

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

Short answer

The determinant of the given matrix is zero because it is an odd-dimension antisymmetric matrix.

Step-by-step solution

Recognize the matrix type

Observe that the given matrix is a square matrix and is 3x3 in size.

Identify the pattern

Notice that the matrix appears to have antisymmetric properties. Specifically, for each element at position (i,j), the element at position (j,i) is its negative.

Confirm antisymmetry

Verify antisymmetry by checking that for all entries of the matrix, a_{ij} = -a_{ji}For example:a_{12} = 2 and a_{21} = -2a_{13} = -3 and a_{31} = 3a_{23} = 4 and a_{32} = -4This confirms that the matrix is antisymmetric.

Use the property of determinants of antisymmetric matrices

Recall that for an n x n antisymmetric matrix, where n is odd, the determinant is always zero. Since the given matrix is 3x3 (n is odd), its determinant must be zero.

Key concepts

antisymmetric matrix

Antisymmetric matrices, also known as skew-symmetric matrices, are square matrices that have a unique property: each element at position \(i,j\) is the negative of the element at position \(j,i\). Formally, a matrix \(A\) is antisymmetric if \(A^T = -A\), where \(A^T\) is the transpose of \(A\). For example, in the given matrix \(\begin{bmatrix}0 & 2 & -3 \ -2 & 0 & 4 \ 3 & -4 & 0\end{bmatrix}\), element \(a_{12} = 2\) and \(a_{21} = -2\), which means \(a_{12} = -a_{21}\). This pattern holds true for all corresponding elements across the diagonals, confirming that the matrix is antisymmetric.
Antisymmetric matrices have some interesting properties that we can use to solve problems, such as finding determinants without direct calculation.

matrix properties

Matrices come with a variety of properties that simplify complex mathematical exercises. For instance, understanding whether a matrix is symmetric, antisymmetric, diagonal, or identity helps in solving linear algebra problems efficiently.
Symmetric matrices are those where \(A_{ij} = A_{ji}\), i.e., they are equal to their transpose. Antisymmetric matrices, as discussed, meet the condition \(A_{ij} = -A_{ji}\).
The given matrix is confirmed to be antisymmetric. Knowing this, we tap into specific properties such as:

• Determinants of antisymmetric matrices of odd order are always zero.
• For even-order antisymmetric matrices, the determinant may or may not be zero.

Recognizing these properties is key to efficiently solving exercises involving determinants and matrix classifications.

zero determinant

The determinant of a matrix is a critical value that provides insights into the properties of the matrix. A zero determinant indicates that the matrix is singular, meaning it does not have an inverse.
For antisymmetric matrices of odd order, such as our 3x3 example, the determinant is always zero. This occurs because the sum of pairs of entries multiplied (one from the original matrix and one from its negative transpose) cancels out, leading to zero. Hence, without any complicated calculations, we can assert that the determinant of our given matrix is zero.
This principle is tremendously useful, saving time and effort in computation.

• It simplifies the process of determining matrix invertibility.
• It aids in solving systems of linear equations by indicating no unique solutions exist if the determinant is zero.

Understanding why the determinant is zero helps students see the bigger picture of matrix behavior and relationships.