Problem 14 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Without actual expansion, prove that: $$ \left|\begin{array}{ccc} 0 & 99 & -998 \\ -99 & 0 & 997 \\ 998 & -997 & 0 \end{array}\right|=0 $$

Step-by-Step Solution

Answer

The determinant is 0 due to the matrix being skew-symmetric with odd dimensions.

Step 1: Recognize the Matrix Type

The given matrix is a 3x3 matrix, specifically, a skew-symmetric matrix. A skew-symmetric matrix is one where the elements satisfy the condition \( A^T = -A \), meaning that the diagonal elements are zero, and each element \( a_{ij} \) is the negative of \( a_{ji} \). This is confirmed by observing that \( a_{12} = 99 \) and \( a_{21} = -99 \), etc.

Step 2: Use Properties of Skew-Symmetric Matrices

An important property of skew-symmetric matrices is that their determinant is zero if the matrix has an odd dimension, e.g., 3x3. This is because such matrices have eigenvalues that occur in pairs of the form \( \lambda \) and \(-\lambda \). In a 3x3 matrix, there must be one eigenvalue that stands alone, and for a skew-symmetric matrix, it can only be zero, implying the determinant is zero.

Key Concepts

Matrix Determinant

The determinant of a matrix is an important concept in linear algebra. It is a scalar value that can be calculated from the elements of a square matrix. Determinants are used in various matrix operations and can provide important insights into the properties of a matrix.

• To find the determinant of a 3x3 matrix, it involves a more complex calculation than for a 2x2 matrix.
• The determinant can indicate whether a matrix is invertible; if the determinant is zero, the matrix doesn't have an inverse.

When dealing with specific types of matrices, such as skew-symmetric matrices, certain properties allow for easier determination of the determinant. Namely, for a skew-symmetric matrix of an odd dimension, the determinant is always zero. This simplifies many calculations and proofs, as seen in our example.

Matrix Properties

Understanding the properties of matrices provides insight into their behavior and the interactions between vectors and linear transformations they represent. For the skew-symmetric matrix given in the exercise:

• A skew-symmetric matrix is any square matrix that is equal to the negative of its transpose ( A^T = -A ).
• This specific structure leads to the property that all diagonal elements are zero.
• Additionally, the off-diagonal elements satisfy equation a_{ij} = -a_{ji} .

These properties are not just theoretical; they have real implications for computations and can simplify finding eigenvalues and determinants. In our example, these properties instantly show that the determinant equals zero, without calculation.

Eigenvalues

Eigenvalues are central to understanding the characteristics of matrices. They provide information about the directions of stretching in linear transformations.

• The eigenvalues of a matrix are scalars that satisfy the equation | A - λI | = 0, where I is the identity matrix.
• Skew-symmetric matrices have the peculiar property of having eigenvalues in pairs: λ and -λ .
• For a matrix with 3 dimensions, there's an odd number of eigenvalues, inevitably resulting in at least one eigenvalue being zero.

Hence, for skew-symmetric matrices with odd dimensions, this zero eigenvalue confirms the determinant is zero, further showcasing how eigenvalues enhance understanding of matrix properties.

3x3 Matrix

The 3x3 matrix is a step up from smaller matrices but provides a manageable size for exploration and computation involving various matrix properties.

• 3x3 matrices are the smallest odd-dimensional matrices that can exhibit interesting properties like skew-symmetry.
• When dealing with a 3x3 skew-symmetric matrix, its simplicity and symmetry allow for clear visualization and understanding of its characteristics, like zero diagonal elements and specific relations between off-diagonal elements.
• They often serve as a practical example size for proofs and demonstrations in linear algebra, offering insights into larger matrices without unnecessary complexity.

By understanding a 3x3 matrix like the given example, we grasp foundational matrix concepts applicable to larger matrices.