Skew-Symmetric Matrix Determinant Explained
A square matrix 'P' is defined as skew-symmetric if its transpose equals its negative. In mathematical terms, this property is expressed as:
\( P^T = -P \)
This definition implies that for any entry \(p_{ij}\) at row \(i\) and column \(j\), the entry \(p_{ji}\) at row \(j\) and column \(i\) must satisfy \(p_{ji} = -p_{ij}\). A direct consequence is that all elements on the main diagonal of a skew-symmetric matrix must be zero. This is because for a diagonal element \(p_{ii}\), the condition becomes \(p_{ii} = -p_{ii}\), which simplifies to \(2p_{ii} = 0\), meaning \(p_{ii} = 0\).
Properties of Skew-Symmetric Matrix Determinants
To determine the value of the determinant of a skew-symmetric matrix, we utilize fundamental properties of determinants:
- The determinant of a matrix is invariant under transposition: \(\det(P^T) = \det(P)\).
- When a matrix is multiplied by a scalar \(k\), its determinant is multiplied by \(k^n\), where \(n\) is the order (size) of the matrix: \(\det(kA) = k^n \det(A)\).
Using the definition \(P^T = -P\), we can equate the determinants:
\( \det(P^T) = \det(-P) \)
Applying the properties mentioned above, we substitute \(\det(P^T)\) with \(\det(P)\) and \(\det(-P)\) with \((-1)^n \det(P)\):
\( \det(P) = (-1)^n \det(P) \)
Applying the Property to Order 3
The specific question concerns a skew-symmetric matrix P of order 3. Thus, the size of the matrix is \(n = 3\). Since 3 is an odd number, the term \((-1)^n\) becomes \((-1)^3\), which equals -1.
Substituting \(n=3\) into the equation \(\det(P) = (-1)^n \det(P)\), we get:
\( \det(P) = (-1)^3 \det(P) \)
\( \det(P) = -1 \cdot \det(P) \)
\( \det(P) = -\det(P) \)
To find the value of \(\det(P)\), we can rearrange the equation:
\( \det(P) + \det(P) = 0 \)
\( 2 \det(P) = 0 \)
Finally, dividing by 2 yields:
\( \det(P) = \frac{0}{2} \)
\( \det(P) = 0 \)
This result demonstrates a key principle: the determinant of any skew-symmetric matrix with an odd order is always zero.
| Matrix Order (n) | Determinant Equation | Result for det(P) |
| Odd (e.g., 3) | \(\det(P) = (-1)^n \det(P)\) implies \(\det(P) = -\det(P)\) | \(\det(P) = 0\) |
| Even (e.g., 2, 4) | \(\det(P) = (-1)^n \det(P)\) implies \(\det(P) = \det(P)\) | Can be non-zero |
Therefore, for the given matrix P, which is skew-symmetric and of order 3, its determinant is 0.
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 :- Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
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?
- 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?
