Problem 16 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement \(-1:\) Determinant of a skew-symmetric matrix of order 3 is zero. Statement - 2 : For any matrix \(\mathrm{A}, \operatorname{det}(\mathrm{A})^{\mathrm{T}}=\operatorname{det}(\mathrm{A})\) and \(\operatorname{det}(-\mathrm{A})=-\operatorname{det}(\mathrm{A})\). Where det (B) denotes the determinant of matrix B. Then: (a) Both statements are true (b) Both statements are false (c) Statement- 1 is false and statement- 2 is true (d) Statement- 1 is true and statement- 2 is false

Step-by-Step Solution

Answer

(d) Statement- 1 is true and statement- 2 is false.

Step 1: Understanding Statement 1

A skew-symmetric matrix is defined such that for any square matrix \( A \), \( A = -A^T \). For a skew-symmetric matrix of odd order, its determinant is always zero because the characteristic polynomial has zero as an eigenvalue an odd number of times (because eigenvalues of skew-symmetric matrices come in pairs of \(\lambda\) and \(-\lambda\)). Therefore, Statement 1 is true.

Step 2: Understanding Part of Statement 2

For any square matrix \( A \), \( \operatorname{det}(A^T) = \operatorname{det}(A) \) is true. This is because the determinant of a matrix remains unchanged when it is transposed. Therefore, the first part of Statement 2 is true.

Step 3: Analyzing the Second Part of Statement 2

Consider \( \operatorname{det}(-A) = -\operatorname{det}(A) \) . This is not generally true; if \( A \) is of odd order, \( \operatorname{det}(-A) \) is indeed \((-1)^n\operatorname{det}(A) \) where \( n \) is the order of the matrix. However, if the matrix order is even, \( (-1)^n = 1 \) and thus the second part of Statement 2, "det(-A) = -det(A)", doesn't hold for even orders. Thus, the statement as given is false.

Step 4: Conclusion Based on Analysis

Based on the analysis: Statement 1 is true and both parts of Statement 2, as presented, do not hold universally. Hence, the correct answer is that Statement 1 is true and Statement 2 is false.

Key Concepts

Skew-Symmetric Matrix

Let's dive into the fascinating world of skew-symmetric matrices. A matrix is termed skew-symmetric if it is equal to the negative of its transpose. In simpler terms, a square matrix \( A \) is skew-symmetric if \( A = -A^T \). This definition results in some interesting properties. For instance:

• All diagonal elements of a skew-symmetric matrix must be zero. This happens because the element at the position \( A_{ii} \) must satisfy \( A_{ii} = -A_{ii} \), which is only possible when \( A_{ii} = 0 \).
• For skew-symmetric matrices of odd order (like a 3x3 matrix), the determinant is always zero. This occurs because the eigenvalues of these matrices are symmetric around zero, and with an odd number of total eigenvalues, at least one must be zero, making the determinant zero.

This distinctive property of skew-symmetric matrices is essential, especially in higher-level mathematics and physics.

Matrix Determinants

Determining the determinant of a matrix is a key operation in linear algebra. The determinant is a scalar value that is a function of a square matrix. It provides insight into various properties of the matrix, such as whether it is invertible or the volume scaling factor of the linear transformation it represents. Here are some important insights:

• For any square matrix \( A \), the determinant can often be interpreted as a scaling factor in transformations. If \( \operatorname{det}(A) = 0 \), the matrix is singular, meaning it does not have an inverse.
• When dealing with a skew-symmetric matrix of odd order, its determinant is always zero, due to the nature of their eigenvalue distribution.
• Furthermore, \( \operatorname{det}(A^T) = \operatorname{det}(A) \). This property shows that transposing a matrix does not affect its determinant.
• In contrast, multiplying a matrix by a scalar \( -1 \) should change its determinant based on the matrix's order, highlighting the importance of understanding these operations distinctively.

Transposition Properties

The transpose operation on a matrix swaps its rows with columns, transforming matrix \( A \) into \( A^T \). This operation retains some vital properties that are widely used in algebraic manipulations:

• For any matrix \( A \), its transpose has the same determinant as the original matrix, \( \operatorname{det}(A^T) = \operatorname{det}(A) \). This consistency makes the determinant a reliable tool irrespective of matrix orientation.
• This property is particularly useful in proving or simplifying equations, as it allows transpositions without worrying about altering the determinant value.
• An intriguing aspect arises when considering \( \operatorname{det}(-A) \). For a matrix of order \( n \), multiplying each element by \( -1 \) gives \( \operatorname{det}(-A) = (-1)^n\operatorname{det}(A) \). For odd \( n \), this results in \( \operatorname{det}(-A) = -\operatorname{det}(A) \), while for even \( n \), \( \operatorname{det}(-A) = \operatorname{det}(A) \).

These properties make transpositions a powerful tool in the analysis and manipulation of matrices.