I. \(|M + M^T| = |M| + |M^T|\) if M is symmetric.
II. \(|M + M^T| = 0\) if M is anti-symmetric.
Select the answer using the code given below :
II only
To answer the question, we need to analyze the properties of symmetric and anti-symmetric matrices, particularly with respect to determinants in a \(2 \times 2\) matrix context.
First, let's understand the definitions:
- A matrix \( M \) is symmetric if \( M = M^T \). This means it is equal to its transpose.
- A matrix \( M \) is anti-symmetric (or skew-symmetric) if \( M = -M^T \). This means its transpose is the negative of the matrix.
Now, let's analyze the options given:
- Assertion I: \( |M + M^T| = |M| + |M^T| \) if \( M \) is symmetric.
Consider a symmetric matrix \( M \). By definition, \( M = M^T \). Therefore, \( M + M^T = M + M = 2M \).
In general, for any square matrix \( A \), the determinant of a scalar multiple is given by:
\(|cA| = c^n |A|\), where \( n \) is the order of the matrix, and \( c \) is a scalar.
Since \( M \) is a \( 2 \times 2 \) matrix, \( n = 2 \). So,
\(|2M| = 2^2 |M| = 4|M|\)
However, since \( M = M^T \),
\(|M^T| = |M|\) (because the determinant of a matrix and its transpose are equal).
Thus, \( |M + M^T| = 4|M| \neq |M| + |M^T| = 2|M| \).
Therefore, Assertion I is incorrect.
- Assertion II: \( |M + M^T| = 0 \) if \( M \) is anti-symmetric.
Consider an anti-symmetric matrix \( M \) such that \( M = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix} \).
In general, \( M + M^T = M + (-M) = 0 \) (the zero matrix) since \(M^T = -M\).
The determinant of the zero matrix, regardless of its dimensions, is always zero.
Therefore, for anti-symmetric matrices, \( |M + M^T| = |0| = 0 \).
Hence, Assertion II is correct.
Conclusion: Based on the analysis, the correct answer is II only. Assertion I is incorrect while Assertion II is correct. The given option, therefore, is II only.
