Problem 74 - Algebra for JEE ADVANCED

Question

If \(A\) is symmetric as well as skew-symmetric matrix, then \(A\) is a. diagonal matrix b. null matrix c. triangular matrix d. none of these

Step-by-Step Solution

Answer

b. null matrix

Step 1: Understanding Symmetric and Skew-Symmetric Matrices

A symmetric matrix satisfies the condition \(A = A^T\), meaning the matrix is equal to its transpose. A skew-symmetric matrix satisfies \(A = -A^T\), meaning the transpose of the matrix is equal to the negative of the matrix.

Step 2: Identifying Common Elements

For a matrix \(A\) to be both symmetric and skew-symmetric, it must satisfy both \(A = A^T\) and \(A = -A^T\) simultaneously. Combine these to obtain the equation \(A = -A\). By adding \(A\) to both sides, we have \(2A = 0\), hence \(A = 0\).

Step 3: Conclusion of the Matrix Type

Since \(A\) is the zero matrix (from Step 2), it doesn't have any non-zero elements and thus is a null matrix. Therefore, it aligns with option b (null matrix).

Key Concepts

Matrix Transpose

The concept of a matrix transpose is central in linear algebra. It involves flipping a matrix over its diagonal, effectively switching the row and column indices of each element in the matrix. So, if you have a matrix \(A = [a_{ij}]\), the transpose, denoted as \(A^T\), would be \([b_{ji}]\), where \(b_{ji} = a_{ij}\). Here are some helpful points to remember:

• If the matrix is symmetric, it's equal to its transpose. Mathematically, this is expressed as \(A = A^T\).
• If a matrix is skew-symmetric, each element is the negative of its transpose: \(A = -A^T\).
• Transposing a matrix twice will yield the original matrix: \((A^T)^T = A\).

Understanding transpose helps in exploring whether a matrix is symmetric or skew-symmetric, and this is vital in solving problems related to matrix equations.

Null Matrix

In linear algebra, a null matrix is one where all its elements are zero. It acts as the additive identity in matrix addition, meaning any matrix added to a null matrix will return the original matrix unchanged. For matrix\(A\), if it's null, then all elements \(a_{ij} = 0\).

• For a matrix \(A\) that is both symmetric and skew-symmetric, solving the equation \(A = -A\) leads to \(2A = 0\), simplifying to \(A = 0\). Hence, it must be a null matrix.
• A null matrix is particularly useful in certain mathematical proofs and can simplify expressions in matrix algebra.

The null matrix concept is often encountered when confirming a matrix that satisfies properties of both symmetry types, as demonstrated in the exercise.

Properties of Matrices

Matrices come with a set of properties that are crucial in various mathematical computations. Understanding these properties helps in matrix operations and transformations. Here's a closer look:

• Symmetric matrices are equal to their transposes \(A = A^T\). Such matrices are often used in statistics and linear transformations.
• Skew-symmetric matrices satisfy \(A = -A^T\), meaning their diagonal elements are always zero.
• The combination of these properties can lead to interesting scenarios; for instance, a matrix that is both symmetric and skew-symmetric must be a null matrix.
• Matrices also exhibit properties like associative, distributive, and commutative laws under specific conditions.

These are foundational properties that must be considered when dealing with complex algebraic operations and when solving matrix-related problems.