Problem 27 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{A}\) and \(\mathrm{B}\) be any two \(3 \times 3\) matrices. If \(\mathrm{A}\) is symmetric and \(\mathrm{B}\) is skewsymmetric, then the matrix \(\mathrm{AB}-\mathrm{BA}\) is: (a) skewsymmetric [Online April 19, 2014] (b) symmetric (c) neither symmetric nor skewsymmetric (d) I or \(-\mathrm{I}\), where \(\mathrm{I}\) is an identity matrix.

Step-by-Step Solution

Answer

(a) skew-symmetric

Step 1: Define the matrix properties

A symmetric matrix, \(\mathbf{A}\), satisfies \(\mathbf{A}^T = \mathbf{A}\). A skew-symmetric matrix, \(\mathbf{B}\), satisfies \(\mathbf{B}^T = -\mathbf{B}\). We need to use these properties to find the nature of the matrix \(\mathbf{AB} - \mathbf{BA}\).

Step 2: Find transpose of \(\mathbf{AB} - \mathbf{BA}\)

Compute the transpose of the expression: \((\mathbf{AB} - \mathbf{BA})^T = (\mathbf{AB})^T - (\mathbf{BA})^T\). Using the properties of transpose, \((\mathbf{AB})^T = \mathbf{B}^T \mathbf{A}^T\) and \((\mathbf{BA})^T = \mathbf{A}^T \mathbf{B}^T\).

Step 3: Substitute properties into the transpose

Apply the properties of \(\mathbf{A}\) and \(\mathbf{B}\): \((\mathbf{AB})^T = \mathbf{B}^T \mathbf{A}^T = -\mathbf{B} \mathbf{A}\) and \((\mathbf{BA})^T = \mathbf{A}^T \mathbf{B}^T = \mathbf{A}(-\mathbf{B}) = -\mathbf{AB}\). Hence, \((\mathbf{AB} - \mathbf{BA})^T = -\mathbf{BA} + \mathbf{AB} = - (\mathbf{AB} - \mathbf{BA})\).

Step 4: Conclude the nature of \(\mathbf{AB} - \mathbf{BA}\)

Since \((\mathbf{AB} - \mathbf{BA})^T = - (\mathbf{AB} - \mathbf{BA})\), the matrix \(\mathbf{AB} - \mathbf{BA}\) is skew-symmetric. Therefore, the correct answer is (a) skew-symmetric.

Key Concepts

Matrix Transpose Properties

The transpose of a matrix involves flipping it over its diagonal. In simpler terms, rows become columns and vice versa. Mathematically, if we have a matrix \( \mathbf{C} \) with elements \( c_{ij} \), then the transpose of this matrix, denoted \( \mathbf{C}^T \), will have elements \( c_{ji} \). This means that every element at position \((i, j)\) in the original matrix is moved to position \((j, i)\) in the transposed matrix. This property is foundational in the study of matrices.
Understanding the transpose is crucial because it helps in identifying other matrix properties, such as those of symmetric and skew-symmetric matrices:

• A symmetric matrix is one where the transpose is equal to the original matrix itself: \( \mathbf{A}^T = \mathbf{A} \).
• A skew-symmetric matrix is one where the transpose is equal to the negative of the original matrix: \( \mathbf{B}^T = -\mathbf{B} \).

Additionally, the transpose of a product of two matrices (\( \mathbf{C} = \mathbf{AB} \)) follows a specific rule: the transpose of the product is the product of the transposes in reverse order, \( (\mathbf{AB})^T = \mathbf{B}^T \mathbf{A}^T \). This property is key when dealing with expressions that involve multiple matrices, such as in our exercise where \( \mathbf{AB} - \mathbf{BA} \) is analyzed.

Matrix Multiplication

Matrix multiplication extends the idea of multiplying numbers with properties and rules that are specific to matrices. In matrix multiplication, each element of the resulting matrix is the sum of products of corresponding elements from rows and columns of the multiplicands. If two matrices \( \mathbf{M} \) and \( \mathbf{N} \) are to be multiplied, the element in the \(i\)-th row and \(j\)-th column of the resulting matrix \( \mathbf{MN} \) is given by the sum \( \sum_{k} m_{ik} n_{kj} \), where the elements used are from the \(i\)-th row of \( \mathbf{M} \) and the \(j\)-th column of \( \mathbf{N} \).
For multiplication to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This is why it is not always possible to multiply any two matrices. Unlike number multiplication, matrix multiplication is not commutative, meaning \( \mathbf{MN} eq \mathbf{NM} \) in general.
In the context of the exercise, this property is crucial: \( \mathbf{AB} \) might not equal \( \mathbf{BA} \), and when combined in an expression like \( \mathbf{AB} - \mathbf{BA} \), the order of multiplication significantly impacts the result.

Matrix Theory

Matrix theory delves into the deeper properties of matrices and their applications. It's a branch of mathematics that focuses on the study of matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns.
Matrices are used widely in various areas such as physics, computer graphics, statistics, and economics. In linear algebra, matrices are particularly important due to their ability to represent linear transformations, solve systems of linear equations, and define spaces and linear mappings.

• Symmetric matrices in theory are crucial as they exhibit properties that simplify a variety of mathematical operations. They often arise in nature and engineering problems and have real eigenvalues and orthogonal eigenvectors.
• Skew-symmetric matrices are interesting because for any skew-symmetric matrix \( \mathbf{B} \), the property \( \mathbf{B}^T = -\mathbf{B} \) ensures that the diagonal elements are zero. Such matrices are particularly common in applied mathematics for defining antisymmetric forms.

Understanding these concepts provides insight into why the interaction of a symmetric matrix \( \mathbf{A} \) and a skew-symmetric matrix \( \mathbf{B} \) produces another skew-symmetric matrix in the form of \( \mathbf{AB} - \mathbf{BA} \). This relates back to the nature of how matrices combine through addition and multiplication, and more broadly to how they align with linear transformations and other mathematical operations.