Which of the following statement is false?a)Every skew symmetric matri...
Every skew symmetric matrix of an odd order is singular or its determinant will be equal to zero.
Which of the following statement is false?a)Every skew symmetric matri...
The statement "Every skew symmetric matrix of odd order is non-singular" is false.
Skew Symmetric Matrix:
A skew symmetric matrix is a square matrix in which the transpose of the matrix is equal to the negative of the matrix itself. In other words, for a skew symmetric matrix A, A^T = -A.
Non-Singular Matrix:
A non-singular matrix is a square matrix that has an inverse. In other words, if A is a non-singular matrix, there exists a matrix B such that AB = BA = I, where I is the identity matrix.
Explanation:
To prove that the statement is false, we need to provide a counterexample.
Counterexample:
Let's consider a 3x3 skew symmetric matrix A as follows:
A = [0 1 0]
[-1 0 0]
[0 0 0]
Now, let's find the determinant of matrix A.
det(A) = (0 * 0 * 0) + (1 * 0 * 0) + (0 * (-1) * 0) - (0 * 0 * (-1)) - (1 * 0 * 0) - (0 * 0 * 0) = 0
Since the determinant of matrix A is zero, we can conclude that matrix A is singular. Therefore, the statement "Every skew symmetric matrix of odd order is non-singular" is false.
In this counterexample, we have shown that a skew symmetric matrix of odd order can indeed be singular.
Conclusion:
The false statement is that every skew symmetric matrix of odd order is non-singular. We have provided a counterexample to demonstrate that this statement does not hold true.