Problem 19 - A Complete Resource Book in Mathematics for JEE Main 2017

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

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

Step 1: Determine Symmetric Matrix Condition

A matrix is symmetric if it is equal to its transpose: that is, a matrix \( C \) is symmetric if \( C = C^T \). This property will be useful to verify the symmetry of \( A(BA) \) and \((AB)A\).

Step 2: Analyze Statement 1: Check Symmetry of \(A(BA)\)

Using the property from Step 1, we first check if \(A(BA)\) is symmetric. \( (A(BA))^T = (BA)^T A^T = A^T (BA)^T = A^T A B^T = ABB = A(BA) \). Thus, \(A(BA)\) is symmetric since it equals its transpose.

Step 3: Analyze Statement 1: Check Symmetry of \((AB)A\)

Similarly, check if \((AB)A\) is symmetric. \(((AB)A)^T = A^T(AB)^T = A^TB^TA^T = ABB = (AB)A\). Hence, \((AB)A\) is symmetric since it equals its transpose.

Step 4: Analyze Statement 2: Commute to Confirm Symmetry of AB

Statement 2 says \(AB\) is symmetric if \(A\) and \(B\) commute. If \(AB = BA\), then \((AB)^T = (BA)^T = A^TB^T = AB\). Hence, \(AB\) is symmetric if \(A\) and \(B\) commute.

Step 5: Determine Matching Option

Both statements are true, but statement 2 explains symmetry in a different context than needed to explain statement 1. Hence, option C is the correct choice, as both are true, but statement 2 is not a correct explanation for statement 1.

Key Concepts

Matrix Transpose

In the world of matrices, the transpose is a simple yet powerful concept. When we transpose a matrix, we flip it over its diagonal. This means the row and column indices swap places. For a matrix \( C \), the transpose is denoted as \( C^T \). If you think of the original matrix as a table, transposing it turns rows into columns, and columns into rows.

For example, if the matrix

• \( C = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \),

then its transpose will be

• \( C^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix} \).

The transpose has an interesting relationship with symmetric matrices, as a matrix is symmetric if it is equal to its transpose, meaning \( C = C^T \). This property simplifies many operations and checks, especially when determining if complex matrix products are symmetric, as in the exercise you encountered.

Understanding transposition gives us critical insights into symmetry and helps in operations like matrix multiplication, where knowing about the transpose can sometimes offer shortcuts or confirmations of properties.

Commutative Property

The commutative property is a concept we often take for granted with numbers. It simply means that you can switch the order of operations, and in most cases, the outcome remains the same. For instance, when we say numbers are commutative under addition, we mean \( a + b = b + a \).

In matrix arithmetic, the commutative property is less common. For matrices \( A \) and \( B \), \( AB \) is generally not the same as \( BA \). However, if \( AB = BA \), we say that \( A \) and \( B \) commute. This becomes a special scenario in matrix multiplication, providing particular results like symmetry.

In your exercise, the commutative property played an important role. Statement 2 asserted that if matrices \( A \) and \( B \) commute, then \( AB \) would be symmetric. This is because the product \( AB \) would be the same as \( BA \), and hence, its transpose \( (AB)^T \) equals \( AB \), confirming symmetry. Understanding when and why matrices commute is crucial for more advanced operations and simplifies determining matrix properties.

Matrix Multiplication

Matrix multiplication, while more complex than scalar multiplication, is a fundamental operation in linear algebra. When you multiply two matrices, you essentially perform a series of dot products between rows and columns of the matrices. This is best visualized by considering a matrix \( A \) with dimensions \( m \times n \) and matrix \( B \) with dimensions \( n \times p \). The resulting matrix \( C = AB \) will have dimensions \( m \times p \).

The interesting thing about matrix multiplication is that it is not commutative. This means \( AB \) is not necessarily equal to \( BA \). The order in which matrices are multiplied often changes the results, a notion which is crucial when working with symmetric matrices.

• If \( AB = BA \), it leads to special cases such as symmetric products, as mentioned in the exercise.
• The results of multiplication can also provide insights into whether a matrix is invertible, identity properties, and much more.

Understanding how to perform and comprehend matrix multiplication is crucial for deeper studies in fields such as machine learning, computer graphics, and physics, where transformations and data manipulation are key.