Problem 50 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option D is correct: all statements are true.

Step 1: Understanding Symmetric and Skew-Symmetric Matrices

A symmetric matrix is one where \( A = A^T \). This means the matrix is equal to its transpose. For skew-symmetric matrices, \( A = -A^T \), meaning the transpose of the matrix is equal to the negative of the matrix itself.

Step 2: Evaluating Option A

If \( A \) is symmetric, then \( A^T = A \). For any positive integer \( n \), using the property that \((A^{n})^T = (A^T)^{n}\), we get \((A^n)^T = (A^T)^n = A^n\). Hence, \( A^n \) is symmetric. Therefore, option A is correct.

Step 3: Evaluating Option B

If \( A \) is skew-symmetric, \( A^T = -A \). For an even \( n \), say \( n = 2k \), we have \((A^n)^T = (A^{2k})^T = ((A^T)^{2k}=((-A)^{2k})). Since (-A)^{2k} = A^{2k}, (A^n)^T = A^n\). Therefore, \( A^n \) is symmetric when \( n \) is even. Option B is correct.

Step 4: Evaluating Option C

If \( A \) is skew-symmetric, \( A^T = -A \). For an odd \( n \), say \( n = 2k+1 \), we have \((A^n)^T = (A^{2k+1})^T = ((A^T)^{2k+1}=((-A)^{2k+1})). Since (-A)^{2k+1} = -A^{2k+1}, (A^n)^T = -A^n\). Therefore, \( A^n \) is skew-symmetric when \( n \) is odd. Option C is correct.

Step 5: Conclusion for Option D

Since all the previous evaluations led to correct conditions, options A, B, and C are all correct. Therefore, option D, which states that all options are correct, is also true.

Key Concepts

Matrix Transpose

When you take the transpose of a matrix, you effectively switch its rows and columns. This operation is fundamental in understanding symmetric and skew-symmetric matrices. If you have a matrix \(A\) and you transpose it to get \(A^T\), then:

• If \(A = A^T\), you have a symmetric matrix.
• If \(A = -A^T\), you get a skew-symmetric matrix.

The matrix transpose operation is crucial in various matrix algebra applications because it helps in checking properties like symmetry. Importantly, the transpose of a transpose brings you back to the original matrix, or mathematically, \((A^T)^T = A\). Understanding transposes allows for deeper insights when exploring concepts like matrix powers, simplifying how we determine whether powers of a matrix retain their symmetry characteristics.

Matrix Powers

Matrix powers involve raising a matrix to a certain integer exponent, much like squaring a number or finding its square root. For any matrix \(A\) and a positive integer \(n\), \(A^n\) represents multiplying the matrix by itself \(n\) times. This concept extends into evaluating properties like symmetry. With symmetric matrices where \(A = A^T\):

• If you raise \(A\) to any power, \(n\), the resulting matrix \(A^n\) will always be symmetric.

For skew-symmetric matrices, it's slightly different:

• If \(n\) is even, \(A^n\) becomes symmetric.
• If \(n\) is odd, \(A^n\) remains skew-symmetric.

Matrix powers simplify working with repeated matrix products and facilitate solving various problems in mathematics and physics, where retaining properties like symmetry is essential.

Matrix Algebra

Matrix algebra is a vast field that covers various operations you can perform on matrices, including addition, multiplication, and finding powers, which tie directly into the concept of symmetricity. When working within this field, understanding the nature of a matrix, such as whether it is symmetric or skew-symmetric, helps determine the outcomes of these operations. Key point in matrix algebra:

• Symmetric and skew-symmetric properties determine how matrices interact when multiplied or added together.
• Addition of two symmetric matrices results in another symmetric matrix.
• Multiplication, however, is more nuanced and depends on the nature of the matrices involved.

Matrix algebra provides a broad toolkit for handling systems of equations, transforming geometrical data, and solving dynamic models in engineering and physics. By understanding these fundamental properties, advanced topics like eigenvectors and eigenvalues become approachable.