Problem 204 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{ll}5 & x \\ y & 0\end{array}\right]\) is a symmetric matrix, then show that \(x=y\).

Step-by-Step Solution

Answer

Based on the property of symmetric matrices, x = y for the given matrix A.

Step 1: Understanding the Symmetric Matrix Concept

A symmetric matrix is a square matrix that is equal to its transpose. Now, the transpose of a matrix is obtained by interchanging the rows and columns. For a matrix A=\[\left[\begin{array}{ll}5 & x \\ y & 0\end{array}\right]\] to be symmetric, A must be equal to its transpose.

Step 2: Derive the Transpose of the Matrix

To transpose a matrix, we swap the row elements to column elements. Therefore, the transpose of the matrix A is A'=\[\left[\begin{array}{ll}5 & y \\ x & 0\end{array}\right]\].

Step 3: Verify the Symmetric Property

According to the property of symmetric matrices, A should be equal to A'. Therefore, we can equate the element at position (1,2) in A to the element at position (1,2) in A', and the element at position (2,1) in A to the element at position (2,1) in A'. Doing this gives us that x = y and y = x, which prove that indeed, x = y.

Key Concepts

Matrix Transpose

Matrix transpose is a fundamental operation of linear algebra. It involves swapping the rows and columns of a matrix. Imagine you have a matrix, similar to a grid of values. By transposing, this grid essentially gets flipped over its diagonal, turning rows into columns.
For instance, consider matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}. \]Its transpose, denoted as \( A^T \), is \[ A^T = \begin{bmatrix} a & c \ b & d \end{bmatrix}. \]
To transpose:

• **Swap elements** across the main diagonal.
• The element in the first row and second column moves to the second row and first column.
• This operation consistently applies to all elements.

Transposing is key to understanding symmetric matrices. If a matrix is equal to its transpose, it is symmetric, revealing deeper properties about its symmetry.

Matrix Equality

Matrix equality is a simple yet crucial concept in understanding matrices. It means two matrices must have the same size and their corresponding elements should be equal.
Consider two matrices:

• \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• \( B = \begin{bmatrix} e & f \ g & h \end{bmatrix} \)

For \( A \) to be equal to \( B \), the following must hold:

• **Size**: Both matrices must be of the same dimension.
• **Elements**: Each corresponding element must be equal, i.e., \( a = e, b = f, c = g, d = h \).

In problems involving symmetric matrices, like in the original exercise, matrix equality is crucial. The symmetry condition implies the matrix alongside its transpose must yield matching elements, particularly symmetric pairs like (1,2) and (2,1), in this case leading to \( x = y \).

Properties of Matrices

Matrices possess several interesting properties that are essential for higher mathematical concepts.
Some noteworthy properties include:

• **Transpose of the Transpose**: Transposing a matrix twice returns the original matrix. Formally, \((A^T)^T = A\).
• **Symmetric Matrices**: A matrix that is equal to its transpose. If \( A = A^T \), then \( A \) is symmetric, as we saw in the exercise, suggesting that off-diagonal elements across the main diagonal are equal.
• **Determinant Property**: The determinant of a square matrix and its transpose is the same: \( \det(A) = \det(A^T) \).
• **Addition and Multiplication**: Matrix addition is both commutative, \( A + B = B + A \), and associative, \((A + B) + C = A + (B + C)\).

Understanding these properties gives deeper insights into matrix behavior and their real-world applications, from computer graphics to physics simulations, making them a central component in linear algebra.