Problem 29 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(A\) and \(B\) be two symmetric matrices of order 3 . [2011] Statement-1: \(A(B A)\) and \((A B) A\) are symmetric matrices. Statement- \(\mathbf{2}: A B\) is symmetric matrix if matrix multiplication of \(A\) with \(B\) 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: Analyze Statement-1

For matrices \(A\) and \(B\) being symmetric, check if \(A(B A)\) is symmetric. A matrix \(X\) is symmetric if \(X^T = X\). Now, \( (A(B A))^T = (A^T B^T A^T) = (A B A) = A(B A) \), showing that it is symmetric. Similarly, for \((A B) A\), \((A B) A)^T = A^T (A B)^T = A (A B) = (A B) A\), hence it is symmetric. Thus, Statement 1 is true.

Step 2: Examine Statement-2

Statement-2 claims that \(A B\) is symmetric if matrix multiplication of \(A\) with \(B\) is commutative. For \(A B\) to be symmetric, \((A B)^T = B^T A^T = A B\). Since \(A\) and \(B\) are symmetric, this simplifies to \(B A = A B\), indicating that commutativity of multiplication indeed implies \(A B\) is symmetric. Thus, Statement 2 is true.

Step 3: Determine Relationship Between Statements

Evaluate if Statement-2 correctly explains Statement-1. In Statement-1, although \(A (B A)\) and \((A B) A\) are symmetric, they are not necessarily symmetric because \(A B\) is symmetric. They are symmetric due to the symmetric properties of \(A\) and \(B\). Therefore, Statement-2 does not explain Statement-1.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation for matrices. When we multiply two matrices, say matrix \(A\) and matrix \(B\), the product matrix is formed by calculating the dot product of the rows of \(A\) with the columns of \(B\). For this to be possible, the number of columns in matrix \(A\) must equal the number of rows in matrix \(B\).

• The resulting matrix will have its dimensions defined by the number of rows from the first matrix and the number of columns from the second matrix.
• This operation is generally not commutative, meaning \(AB eq BA\) in most cases.
• However, when both matrices are symmetric and multiplication is commutative, \(AB = BA\), creating interesting properties.

Understanding these basics helps us see why matrix multiplication works the way it does. In our specific exercise, this non-commutativity is key in understanding the interaction of symmetric matrices \(A\) and \(B\).

Commutative Property

The commutative property is familiar from basic arithmetic where numbers can be added or multiplied in any order without affecting the result. For matrices, however, this property does not generally apply to multiplication. Normally, \(AB eq BA\) because matrix multiplication involves the alignment and summation of dots and slices from different dimensions.

• Only in special cases, such as with certain pairs of symmetric matrices, do we see these products equal.
• In our exercise, if two symmetric matrices \(A\) and \(B\) commute, then together they perform as one entity, leading \(AB = BA\), which was needed to confirm statement properties about symmetry.

The exercise provides a critical glimpse into how symmetric properties of matrices can lead to commutative-like behavior, underpinning the deeper connections in linear algebra.

Transpose of a Matrix

The transpose of a matrix is a simple yet powerful concept in linear algebra. Transposing a matrix means flipping it over its diagonal, swapping the row and column indices: if \(A\) is a matrix, then \(A^T\) is obtained by converting row \(i\) to column \(i\).

• This operation plays a crucial role in determining the symmetry of a matrix.
• For symmetric matrices, \(A^T = A\), so they remain unchanged under transposition.
• In the solution, transposition helped us verify the symmetry of matrix products like \(A(BA)\) and \((AB)A\).

The property that \(A^T = A\) for symmetric matrices also feeds into checking equations to ensure products stay symmetric, capitalizing on the harmony of geometric orientations embodied in these matrices. Understanding the transpose not only helps in symmetry but also in broader linear transformations.