Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

Which of the following is correct? (A) If \(A\) is a symmetric matrix, then \(A^{n}\) is symmetric, \(n \in N\) (B) If \(A\) is a skew-symmetric matrix then \(A^{n}\) is symmetric if \(n\) is even, \(\overline{n \in N}\) (C) If \(A\) is a skew-symmetric matrix then \(A^{n}\) is skew-symmetric if \(n\) is odd, \(n \in N\) (D) All of these

Step-by-Step Solution

Answer

(D) All of these are correct.

Step 1: Understanding Symmetric and Skew-Symmetric Matrices

A symmetric matrix is a square matrix that is equal to its transpose, i.e. \(A = A^T\). A skew-symmetric matrix (or anti-symmetric matrix) is one where the transpose of the matrix equals the negative of the matrix itself, i.e. \(A^T = -A\). Analyze these definitions as we work through the options.

Step 2: Analyzing Option (A)

If \(A\) is symmetric, this means \(A = A^T\). To prove if \(A^n\) is symmetric, we need \((A^n)^T = A^n\). Based on the property of transposes \((A^n)^T = (A^T)^n = A^n\), using \(A = A^T\). Therefore, option (A) is correct as the power of a symmetric matrix is always symmetric.

Step 3: Analyzing Option (B)

For a skew-symmetric matrix \(A^T = -A\), if \(n\) is even, \((A^n)^T = (A^T)^n = (-A)^n = A^n\). Thus for any even power \(n\), \(A^n\) is symmetric. Therefore, option (B) is correct.

Step 4: Analyzing Option (C)

For a skew-symmetric matrix \(A^T = -A\), if \(n\) is odd, \((A^n)^T = (A^T)^n = (-A)^n = -A^n\). Thus for any odd power \(n\), \(A^n\) maintains the skew-symmetry. Therefore, option (C) is correct.

Step 5: Conclusion on Option (D)

Given the correctness of options (A), (B), and (C) from above analysis, all statements are correct. Therefore, option (D) is indeed correct.

Key Concepts

Symmetric Matrix

In linear algebra, a symmetric matrix is a special type of square matrix. A matrix is considered symmetric if it is equal to its transpose. This means that the matrix does not change when it is flipped over its main diagonal. Mathematically, a matrix \(A\) is symmetric if \(A = A^T\).

Consider the properties of symmetric matrices:

• All diagonal elements need to be equal to themselves on the opposite side of the main diagonal.
• For example, a 3x3 symmetric matrix looks like: \[\begin{bmatrix}a & b & c \b & d & e \c & e & f\end{bmatrix}\]
• The power of a symmetric matrix is also symmetric. If you multiply a symmetric matrix by itself \(n\) times, the result will still be symmetric. This is because the transpose of a power of a matrix \( (A^n)^T \) is the same as \(A^n\) when \(A = A^T\).

Symmetric matrices frequently appear in real-world problems, especially in physics and statistics, where symmetry is a natural property of the systems being modeled.

Skew-Symmetric Matrix

A skew-symmetric matrix, also referred to as an anti-symmetric matrix, is similar but has a twist. For a matrix to be skew-symmetric, its transpose must be equal to the negative of the matrix, that is, \(A^T = -A\). This means that each element in the matrix on one side of the diagonal should be the negative of the element on the opposite side.

Key characteristics of skew-symmetric matrices include:

• The elements on the main diagonal must all be zero because they must equal their own negative \(-a = a \Rightarrow a = 0\).
• For example, a 3x3 skew-symmetric matrix might appear as:\[\begin{bmatrix}0 & -b & -c \b & 0 & -e \c & e & 0\end{bmatrix}\]
• When a skew-symmetric matrix is raised to an even power \(n\), it results in a symmetric matrix. Conversely, when it is raised to an odd power, it maintains its skew-symmetry.

These matrices are particularly useful in computational mathematics and engineering fields, where complex dynamics are often decomposed into symmetrical and anti-symmetrical parts.

Matrix Transpose

The transpose of a matrix is a basic operation in matrix algebra. When you transpose a matrix, you turn its rows into columns and its columns into rows. This is essential in the definition of both symmetric and skew-symmetric matrices.

Here's a step-by-step look at how to transpose a matrix:

• Take a matrix \(A\), for instance,\[\begin{bmatrix}1 & 2 & 3 \4 & 5 & 6\end{bmatrix}\]
• Switch its rows with its columns:\[\begin{bmatrix}1 & 4 \2 & 5 \3 & 6\end{bmatrix}= A^T\]
• The element in the \(i^{th}\) row and \(j^{th}\) column of the original matrix becomes the element in the \(j^{th}\) row and \(i^{th}\) column of its transpose.

Understanding the transpose operation is crucial as it plays a vital role in the properties and classification of matrices, especially when checking for symmetry and skew-symmetry, or when dealing with matrix equations.