Problem 73 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two symmetric matrices of order \(3 .\) \([2011]\) Statement 1: \(A(B A)\) and \((A B) A\) are symmetric matrices. Statement 2: \(A B\) is symmetric matrix if matrix multiplication of \(A\) and \(B\) is commutative. (A) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (B) Statement 1 is true, Statement 2 is false. (C) Statement 1 is false, Statement 2 is true. (D) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1

Step-by-Step Solution

Answer

(A) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

Step 1: Understanding Symmetric Matrices

A symmetric matrix is one that is equal to its transpose. So, matrix \( A \) is symmetric if \( A = A^T \), and similarly for \( B \).

Step 2: Evaluate Statement 1: Symmetric Matrix Multiplication Rule

For the product \( A(BA) \) to be symmetric, \( (A(BA))^T = A(BA) \) must be true. Similarly, \( ((AB)A)^T = (AB)A \). We will take the transpose of both products and check if they are equal to the respective products.

Step 3: Calculation for \( A(BA) \)

Since \( A \) and \( B \) are symmetric, we first note that \( (BA)^T = A^T B^T = AB \). So, \( (A(BA))^T = (BA)^T A^T = (AB)A = A(BA) \), confirming \( A(BA) \) is symmetric.

Step 4: Calculation for \( (AB)A \)

Similarly, \( ((AB)A)^T = A^T (AB)^T = A(BA) = (AB)A \), confirming \( (AB)A \) is symmetric as well.

Step 5: Evaluate Statement 2: Commutativity and Symmetry

For \( AB \) to be symmetric, \( (AB)^T = AB \) needs to hold. We have \( (AB)^T = B^T A^T = BA \). Thus, \( AB = BA \) implies commutativity is necessary for \( AB \) to be symmetric.

Step 6: Conclusion from Evaluations

Statement 1 is true because both \( A(BA) \) and \( (AB)A \) were shown to be symmetric using transpose properties. Statement 2 is also true since commutativity is required for \( AB \) to be symmetric.

Key Concepts

Matrix Multiplication

Matrix multiplication is a critical operation in linear algebra. In contrast to numbers, multiplying matrices involves rows and columns to create a new matrix. The product of two matrices, say \( A \) and \( B \), is a matrix \( C \) where each element is computed by taking the row of matrix \( A \) and the column of matrix \( B \) and performing a dot product. This operation is only possible if the number of columns in the first matrix matches the number of rows in the second matrix. Hence, the order of matrices is crucial; more on that later.

• To compute the element in the first row and first column of \( C \), take the first row of \( A \) and multiply it with the first column of \( B \).
• Each element in the resulting matrix is a sum of products, which is why it's referred to as a matrix product.

Understanding matrix multiplication is essential for validating properties like symmetry or finding the product's dimensions.

Commutative Property

The commutative property is a fundamental rule in algebra that states that the order in which two numbers are added or multiplied does not affect the result. However, in matrices, this property does not generally apply. This means that, for matrices \( A \) and \( B \), \( AB \) may not equal \( BA \).

• The commutative property applies to some special matrices, like scalar matrices, identity matrices, or when matrices are both symmetric and commute.
• In most typical applications, care must be taken to maintain the correct order of operations due to the non-commutative nature of matrix multiplication.

The lack of commutativity in general matrix multiplication is why the order of multiplication matters significantly, influencing both the resultant matrix and its properties.

Matrix Transpose

A matrix transpose is an operation that flips a matrix over its diagonal. The rows of the original matrix become the columns in the new matrix and vice versa. If you have a matrix \( A \), its transpose is denoted as \( A^T \).

• The transpose of a 3x2 matrix is a 2x3 matrix.
• If \( A \) is symmetric, then \( A = A^T \). This property holds true for both real and complex matrices as long as the complex conjugate is used in the complex matrix case.

Transposing plays a vital role in determining the properties of matrices, like checking for symmetry, and is useful in many applications across linear algebra.

Order of Matrix

The order of a matrix, often referred to as its dimensions, indicates the number of rows and columns a matrix contains. For instance, a matrix of order 3x3 has 3 rows and 3 columns. Understanding this concept is essential in matrix operations.

• It determines which operations are possible, such as which matrices can be multiplied.
• It aids in understanding how properties, like symmetries, are applied across different matrices.
• The square matrices (equal number of rows and columns) are of particular interest, especially in the study of properties like symmetry and determinant.

Each matrix's order defines the framework in which it functions, directly impacting calculations and conceptual understanding in linear algebra.

Matrices in Linear Algebra

Matrices are foundational to the study of linear algebra and are used to solve systems of linear equations, perform transformations, and more. They are rectangular arrays consisting of numbers or expressions arranged in rows and columns.

• Used to represent and solve linear systems efficiently through methods like Gaussian elimination.
• Key to understanding various algebraic structures, simplifying complex transformations in space.
• They provide a way to work concisely with large data sets and complex operations.

Overall, matrices serve as the backbone of much of algebraic computation, so grasping their diverse properties and operations is crucial for any student delving into linear algebra concepts.