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.
Important Questions from Matrices and Determinants
What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?
In obtaining the solution of the system of equations $x + y + z = 7$, $x + 2y+3z = 16$ and $x + 3y+4z = 22$ by Cramer's rule, the value of $y$ is obtained by dividing $D$ by $D_2$, where
$D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix}$
What is the value of the determinant $D_2$?
Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB)^{-1} = A^{-1}B^{-1}$
II. $(BA)(AB)^{-1} = I$, where $I$ is the identity matrix
III. $(AB)^T = A^T B^T$
How many of the above are correct?
Consider the following statements :
Statement-I :
If $X$ is an $n \times n$ matrix, then $\det(mX) = m^n \det(X)$, where $m$ is a scalar.
Statement-II :
If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det(Y) = m\det(X)$.
Which one of the following is correct in respect of the above statements?
Consider the following statements about the matrix $M = \begin{vmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{vmatrix}$
Statement-I : The inverse of $M$ does not exist.
Statement-II : $M$ is non-singular.
Which one of the following is correct in respect of the above statements?
