Problem 208 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If for a square matrix \(A=\left[a_{i j}\right] a_{i j}=i^{2}-j^{2}\), then show that \(A\) is a skew-symmetric matrix.

Step-by-Step Solution

Answer

Yes, the given matrix \(A\) is skew-symmetric because its transpose \(-A^T\) equals \(A\).

Step 1: Define the given Matrix

Let's define the given matrix \(A\) where each element of the matrix \(a_{i j} = i^2 - j^2\). The matrix will be a square matrix as its number of rows is equal to the number of columns.

Step 2: Transpose of Matrix A

The next step is to find the transpose of the original matrix \(A\). The transpose of a matrix is found by interchanging its rows into columns or vice versa. If the elements of A are \(A = [a_{ij}]\), then the matrix transpose \(A^T\) will have elements \(A^T = [a'_{ji}]\). Since \(a_{ij} = i^2 - j^2\), we will get \(a'_{ji} = j^2 - i^2\).

Step 3: Verify if A is skew-symmetric

The matrix \(A\) will be skew-symmetric if \(A = -A^T\). We can see that by comparing the elements of \(A\) and \(-A^T\). The element \(a_{ij}\) in \(A\) is \(i^2 - j^2\) and the element in \(-A^T\) (in the same position) is \(-j^2 + i^2\). Therefore, A = -A^T and A is a skew-symmetric matrix.

Key Concepts

Matrix Transposition

Matrix transposition is a fundamental operation in linear algebra. It involves flipping a matrix over its diagonal, transforming the rows into columns or vice versa. Transposing a matrix is denoted by the superscript 'T'. For a matrix \(A = [a_{ij}]\), its transpose \(A^T\) will have elements \([a_{ji}]\). This means that for every element \(a_{ij}\) in the original matrix, the corresponding element in the transpose is found at \(a_{ji}\).
This operation is especially important for understanding matrix properties and structures, such as symmetry and skew-symmetry.
In practical terms, matrix transposition is:

• Swapping the row and column indices of each element.
• Maintaining the matrix's dimensions, keeping it a square if it started square.
• Essential for certain matrix identities and transformations.

Mastering matrix transposition offers a solid foundation for exploring more complex algebraic concepts and is crucial for solving many mathematical problems.

Square Matrices

Square matrices are matrices that have the same number of rows and columns. This structure means they form an \(n \times n\) grid, where \(n\) is the number of rows or columns. Square matrices are key elements in linear algebra and have unique properties that differentiate them from other matrices.
Square matrices are essential because:

• They can have determinants and inverses, unlike non-square matrices.
• They enable the definition of matrix operations like trace (sum of diagonal elements).
• They are necessary for defining concepts like eigenvalues and eigenvectors.

In the context of skew-symmetric matrices, like the one in our original exercise, square matrices allow for the property \(A = -A^T\), meaning the matrix equals the negative of its transpose. Understanding square matrices is crucial to fully grasp other advanced matrix concepts.

Matrix Properties

Matrices exhibit a variety of properties that are vital in matrix operations and applications. Understanding these properties helps in analyzing and solving matrix-related problems. Some important properties include:

• Symmetric Matrices: These are matrices that satisfy \(A = A^T\), meaning the matrix is equal to its transpose.
• Skew-Symmetric Matrices: These matrices satisfy the property \(A = -A^T\). Each element \(a_{ij}\) negates as \(-a_{ji}\) upon transposition, which is demonstrated beautifully with the given exercise where \(a_{ij} = i^2 - j^2\).
• Diagonal Dominance: Matrices where diagonal elements are larger than the sum of the absolute values of other elements in the row, useful in numerical analysis.

These properties have significant mathematical implications and applications in various fields, from computer science to engineering, fundamentally aiding in complex problem-solving and data analysis.