I. If \(n \times n\) (\(n > 1\)) matrix is symmetric, then its inverse is also a symmetric matrix.
II. If \(n \times n\) (\(n > 1\)) matrix is singular, then its adjoint is also a singular matrix.
Which of the statements given above is/are correct ?
Statement I Analysis: Symmetric Matrix and its Inverse
A matrix \(A\) is symmetric if its transpose equals the matrix itself, i.e., \(A^T = A\). The inverse \(A^{-1}\) exists only if \(A\) is non-singular (\(det(A) \neq 0\)).
For \(A^{-1}\) to be symmetric, its transpose must equal itself: \((A^{-1})^T = A^{-1}\).
Using the property that the transpose of an inverse is the inverse of the transpose, we have \((A^{-1})^T = (A^T)^{-1}\).
Since \(A\) is symmetric (\(A^T = A\)), we can substitute \(A\) for \(A^T\): \((A^{-1})^T = (A)^{-1} = A^{-1}\).
Thus, if a symmetric matrix \(A\) is non-singular, its inverse \(A^{-1}\) is also symmetric. Statement I is correct.
Statement II Analysis: Singular Matrix and its Adjoint
A matrix \(A\) is singular if its determinant is zero, \(det(A) = 0\). We are given \(n > 1\).
The fundamental relationship between a matrix, its adjoint, and its determinant is:
\(A \cdot adj(A) = det(A) \cdot I\)
If \(A\) is singular (\(det(A) = 0\)), this equation becomes:
\(A \cdot adj(A) = 0 \cdot I = O\)
(where \(O\) is the zero matrix).
To determine if \(adj(A)\) is singular, we examine its determinant. The property relating the determinant of the adjoint to the determinant of the original matrix is:
\(det(adj(A)) = (det(A))^{n-1}\)
Since \(det(A) = 0\) and \(n > 1\) (meaning \(n-1 \ge 1\)), we have:
\(det(adj(A)) = (0)^{n-1} = 0\)
Because the determinant of \(adj(A)\) is 0, the adjoint matrix \(adj(A)\) is singular. Statement II is correct.
Conclusion
Both Statement I and Statement II are correct.
Similar Questions
- Let X be a matrix of order \(3 \times 3\), Y be a matrix of order \(2 \times 3\) and Z be a matrix of order \(3 \times 2\). Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below. Let \(A\) and \(B\) be matrices of order \(3 \times 3\)., If \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\), then what is the value of \(|2B(\text{adj}(3A))|\)?
- Consider the following statements in respect of two non-singular matrices \(A\) and \(B\) of the same order \(n\):
1. \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\)
2. \(\text{adj}(AB) = \text{adj}(BA)\)
3. \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix of order \(n\)
How many of the above statements are correct? - Consider the following statements in respect of a non-singular matrix \(A\) of order \(n\):
1. \(A(\text{adj}A^T) = A(\text{adj}A)^T\)
2. If \(A^2 = A\), then \(A\) is identity matrix of order \(n\)
3. If \(A^3 = A\), then \(A\) is identity matrix of order \(n\)
Which of the statements given above are correct? - Consider the following statements in respect of a skew-symmetric matrix \(A\) of order \(3\):
1. All diagonal elements are zero.
2. The sum of all the diagonal elements of the matrix is zero.
3. \(A\) is orthogonal matrix.
Which of the statements given above are correct? If $A = \begin{bmatrix} \sin 2\theta & 2 \sin^2 \theta - 1 & 0 \\ \cos 2\theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then which of the following statements is/are correct?
1. $A^{-1} = \text{adj}A$
2. $A$ is skew-symmetric matrix
3. $A^{-1} = A^T$
Select the correct answer using the code given below :If
\(\begin{vmatrix}1 + \omega & 1 + \omega^2 & \omega + \omega^2 \\1 & \omega & \omega^2 \\\frac{1}{\omega} & \frac{1}{\omega^2} & 1\end{vmatrix}\)
\(\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
then what isequal to?
- If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
- What is \(A(\text{adj } A)\) equal to?
- If \(\begin{vmatrix} a-b & p-q & x-y \\ b-c & q-r & y-z \\ c-a & r-p & z-x \end{vmatrix} = k \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}\) then what is the value of k ?
Important Questions from Matrices and Determinants
- If each element of a matrix is zero, then the matrix is:
- Let X be a matrix of order \(3 \times 3\), Y be a matrix of order \(2 \times 3\) and Z be a matrix of order \(3 \times 2\). Which of the following statements are correct?
I. (ZY)X is defined and is a square matrix of order 3.
II. Y(XZ) is defined and is a square matrix of order 2.
III. X(YZ) is not defined.
Select the answer using the code given below. Let \(A\) and \(B\) be matrices of order \(3 \times 3\)., If \(|A| = \frac{1}{2\sqrt{2}}\) and \(|B| = \frac{1}{729}\), then what is the value of \(|2B(\text{adj}(3A))|\)?
- Consider the following statements in respect of two non-singular matrices \(A\) and \(B\) of the same order \(n\):
1. \(\text{adj}(AB) = (\text{adj}A)(\text{adj}B)\)
2. \(\text{adj}(AB) = \text{adj}(BA)\)
3. \((AB)\text{adj}(AB) - |AB|I_n\) is a null matrix of order \(n\)
How many of the above statements are correct? - Consider the following statements in respect of a non-singular matrix \(A\) of order \(n\):
1. \(A(\text{adj}A^T) = A(\text{adj}A)^T\)
2. If \(A^2 = A\), then \(A\) is identity matrix of order \(n\)
3. If \(A^3 = A\), then \(A\) is identity matrix of order \(n\)
Which of the statements given above are correct?
