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?
Understanding Skew-Symmetric Matrices
A skew-symmetric matrix is a square matrix \(A\) where its transpose (\(A^T\)) is equal to its negative (\(-A\)). This relationship can be written as:
\(A^T = -A\)
This definition implies that for any element \(a_{ij}\) (the element in the \(i\)-th row and \(j\)-th column), the element \(a_{ji}\) must satisfy \(a_{ji} = -a_{ij}\). We need to check three statements regarding a \(3 \times 3\) skew-symmetric matrix.
Statement 1: Diagonal Elements are Zero
Statement 1 says that all diagonal elements of the matrix are zero.
For any diagonal element, the row index is the same as the column index, meaning \(i=j\). So, we are looking at elements \(a_{ii}\).
Using the property \(a_{ji} = -a_{ij}\) for \(i=j\), we get:
\(a_{ii} = -a_{ii}\)
If we add \(a_{ii}\) to both sides of the equation, we get:
\(2a_{ii} = 0\)
Dividing by 2 yields:
\(a_{ii} = 0\)
This confirms that all diagonal elements (\(a_{11}, a_{22}, a_{33}\)) must be zero. So, Statement 1 is correct.
Statement 2: Sum of Diagonal Elements
Statement 2 claims that the sum of all the diagonal elements is zero.
The sum of the diagonal elements of a matrix is called its trace. For a \(3 \times 3\) matrix \(A\), the trace is \(tr(A) = a_{11} + a_{22} + a_{33}\).
From Statement 1, we established that \(a_{11} = 0\), \(a_{22} = 0\), and \(a_{33} = 0\). Therefore, the sum is:
\(tr(A) = 0 + 0 + 0 = 0\)
This means the sum of the diagonal elements is zero. So, Statement 2 is correct.
Statement 3: Orthogonal Matrix Property
Statement 3 proposes that the skew-symmetric matrix \(A\) is an orthogonal matrix.
A matrix is orthogonal if its transpose equals its inverse, satisfying \(A^T A = I\), where \(I\) is the identity matrix.
For a skew-symmetric matrix, we know \(A^T = -A\). Substituting this into the orthogonality condition:
\((-A)A = I\)
\(-A^2 = I\)
\(A^2 = -I\)
Let's consider a general \(3 \times 3\) skew-symmetric matrix:
| \(A = \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix}\) |
Calculating \(A^2\):
\(A^2 = A \times A = \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix}\)
\(A^2 = \begin{pmatrix} -a^2-b^2 & -bc & ac \\ -bc & -a^2-c^2 & -ab \\ ac & -ab & -b^2-c^2 \end{pmatrix}\)
For \(A\) to be orthogonal, \(A^2\) must equal \(-I\):
\(\begin{pmatrix} -a^2-b^2 & -bc & ac \\ -bc & -a^2-c^2 & -ab \\ ac & -ab & -b^2-c^2 \end{pmatrix} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}\)
This requires the off-diagonal terms to be zero (\(ac=0, -bc=0, ab=0\)) and the diagonal terms to be -1 (\(-a^2-b^2 = -1\), \(-a^2-c^2 = -1\), \(-b^2-c^2 = -1\)).
If \(a=0\), we need \(b^2=1\), \(c^2=1\), and \(b^2+c^2 = 1\). But \(1+1 = 2 \neq 1\). This shows a contradiction. A general skew-symmetric matrix does not satisfy \(A^2 = -I\). For example, the zero matrix is skew-symmetric but \(0^2 \neq -I\). Therefore, Statement 3 is incorrect.
Final Conclusion
Reviewing the analysis:
- Statement 1 is correct.
- Statement 2 is correct.
- Statement 3 is incorrect.
Thus, the correct statements are 1 and 2 only.
Similar Questions
- If \(M_k = \begin{bmatrix} k & k-1 \\ k-1 & k \end{bmatrix}\) where k is a natural number, then what is \(|M_1| + |M_2| + |M_3| + \dots + |M_{50}|\) equal to ?
- Let \(f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}\) where q is any constant, then what is \(\frac{d^2}{dx^2}(f(x))\) at \(x = 0\) equal to ?
- If p, q, r are the cube roots of unity, then what is \(\begin{vmatrix} p^2+q^2 & r^2 & r^2 \\ p^2 & q^2+r^2 & p^2 \\ q^2 & q^2 & r^2+p^2 \end{vmatrix}\) equal to ?
- If A is a square matrix such that \(|A| = -2\), then \(|AA^T|\), where \(A^T\) is the transpose of A, is equal to
- Consider the following statements :
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 ? - If \(M = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}\), then what is the value of \(|M| |adjM|\) ?
- If \(M^T\) is the transpose of a \(2 \times 2\) matrix M, then which of the following is/are correct ?
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 : - If M is a square matrix such that \(M^3 = M\), then how many values of \(|M|\) are possible ?
- Let \(p = (x + y + z)\) and \(q = xyz\). If \(\begin{vmatrix} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{vmatrix}\) is positive, then which one of the following is correct ?
- 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.
Important Questions from Matrices and Determinants
- If each element of a matrix is zero, then the matrix is:
- If \(M_k = \begin{bmatrix} k & k-1 \\ k-1 & k \end{bmatrix}\) where k is a natural number, then what is \(|M_1| + |M_2| + |M_3| + \dots + |M_{50}|\) equal to ?
- Let \(f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}\) where q is any constant, then what is \(\frac{d^2}{dx^2}(f(x))\) at \(x = 0\) equal to ?
- If p, q, r are the cube roots of unity, then what is \(\begin{vmatrix} p^2+q^2 & r^2 & r^2 \\ p^2 & q^2+r^2 & p^2 \\ q^2 & q^2 & r^2+p^2 \end{vmatrix}\) equal to ?
- If A is a square matrix such that \(|A| = -2\), then \(|AA^T|\), where \(A^T\) is the transpose of A, is equal to
