Problem 19 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(A\) and \(B\) be two symmetric matrices of order 3 . Statement 1: \(A(B A)\) and \((A B) A\) are symmetric matrices. Statement \(2: A B\) is symmetric matrix if matrix multiplication of \(A\) with \(B\) is commutative. (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 if false

Step-by-Step Solution

Answer

Choice (C): Both statements are true; statement 2 does not explain statement 1.

Step 1: Understanding Symmetric Matrices

A symmetric matrix is equal to its transpose, meaning if \( C \) is a symmetric matrix, then \( C = C^T \). This property will be crucial in verifying the statements.

Step 2: Assessing Statement 1

To check if \( A(BA) \) and \( (AB)A \) are symmetric, note that \( A \) and \( B \) being symmetric implies \( A = A^T \) and \( B = B^T \). For \( A(BA) \), we have:\[(A(BA))^T = A^T B^T A^T = ABA = A(BA)\]The expression is symmetric. Similarly, for \((AB)A\),\[((AB)A)^T = A^T B^T A^T = ABA = (AB)A\]Thus, both \(A(BA)\) and \( (AB)A \) are indeed symmetric.

Step 3: Verifying Statement 2

For \( AB \) to be symmetric, it must satisfy \( (AB)^T = AB\). This holds true when \( A \) and \( B \) commute, meaning \( AB = BA\). Thus, Statement 2 is correct, as commutativity of \( A \) and \( B \) guarantees the symmetry of \( AB \).

Step 4: Analyzing Connection Between Statements

Statement 1 being true is not dependent on the correctness of Statement 2. The symmetry in Statement 1 is due to the symmetric nature of expressions independently, not because \( AB \) is symmetric. Hence, Statement 2, although correct, does not explain Statement 1.

Key Concepts

Matrix Multiplication

Matrix multiplication is a crucial operation in linear algebra. Unlike regular multiplication of numbers, the multiplication of matrices is more involved.
To multiply two matrices, you need to follow certain rules. Specifically for matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This condition is crucial for the product to exist.
The resulting matrix, known as the product matrix, has dimensions determined by the number of rows in the first matrix and the number of columns in the second matrix.

• For example, if matrix A is of order \( m \times n \) and matrix B is of order \( n \times p \), then the product matrix will be of order \( m \times p \).

Understanding this concept and applying it correctly can simplify complex problems involving matrices.

Commutative Property

In mathematics, the commutative property is the ability to change the order of operands without changing the result. However, in the context of matrices, this property does not always hold.
For general matrices, the commutative property of multiplication does not apply, which means that \( AB eq BA \) in most cases.
This is important when working with symmetric matrices, because if two symmetric matrices do commute, meaning \( AB = BA \), then their product is also symmetric.

• Checking for commutativity can help determine additional relationships and properties between matrices.
• Thus, for matrix multiplication to result in a symmetric matrix, commutativity is a key requirement.

Understanding when matrices can commute is crucial when exploring their deeper algebraic properties.

Transpose of a Matrix

The transpose of a matrix is a simple yet important transformation. It involves swapping the rows and columns of a matrix.
Denoted as \( A^T \) for a matrix \( A \), the transpose operation plays a pivotal role in verifying symmetry among matrices.

• For any matrix \( A \) with elements \( a_{ij} \), the transpose matrix \( A^T \) will have elements \( a_{ji} \).
• When a matrix is equal to its transpose, it is classified as a symmetric matrix.

Symmetric matrices possess interesting mathematical properties that are often exploited in various applications, such as physics, computer graphics, and optimization.

Order of Matrices

The order of a matrix refers to its dimensions, specifically the number of rows and columns it contains, expressed as \( m \times n \).
Understanding the order of matrices is crucial before performing operations like multiplication, addition, or inversion.
The order is essential in determining if matrix multiplication is feasible, as multiplication rules require the inner dimensions to be equal.

• For example, a matrix of order \( 3 \times 2 \) cannot be multiplied directly with a matrix of order \( 4 \times 3 \).
• Always ensure the dimensions align to carry out successful operations.

Getting comfortable with the concept of matrix order will streamline many matrix-related computations and help avoid common pitfalls.