Matrix [A]nxm is a skew symmetric matrix then, which of the following statements is/are correct ?a)A2n is a skew symmetric matrixb)A2n+1 is a symmetric matrixc)All positive odd integral power of a skew symmetric matrix are skew symmetric matrix and positive even integral powers of a skew symmetric matrix are symmetric matrix.d)The eigen value of skew – symmetric matrix are either purely imaginary or zero.Correct answer is option 'C,D'. Can you explain this answer?

Question

Matrix [A]nxm is a skew symmetric matrix then, which of the following statements is/are correct ?a)A2n is a skew symmetric matrixb)A2n+1 is a symmetric matrixc)All positive odd integral power of a skew symmetric matrix are skew symmetric matrix and positive even integral powers of a skew symmetric matrix are symmetric matrix.d)The eigen value of skew &ndash; symmetric matrix are either purely imaginary or zero.Correct answer is option 'C,D'. Can you explain this answer?

Answer

If a matrix [A]_nxm is a skew-symmetric matrix then,

1. A²ⁿis a symmetric matrix

2. A²ⁿ⁺¹is a skew – symmetric matrix

Answer

Statement a: A^2n is a skew symmetric matrix
To determine if A^2n is a skew symmetric matrix, we need to understand the properties of a skew symmetric matrix. A skew symmetric matrix A is defined as A = -A^T, where A^T is the transpose of matrix A.

When we square a skew symmetric matrix A, we get (A^2) = (-A^T)(-A^T) = (A^T)(A^T). Now, let's take the transpose of (A^2):

((A^2)^T) = (A^T)(A^T)

Since (A^2)^T = (A^2), for A^2n to be a skew symmetric matrix, (A^2n)^T = (A^2n) must hold true.

((A^2n)^T) = ((A^2)^n)^T = ((A^T)(A^T))^n = (A^T)^n(A^T)^n = (A^2)^n

Therefore, (A^2n) = (A^2)^n, which implies that A^2n is not necessarily a skew symmetric matrix. Hence, statement a is incorrect.

Statement b: A^2n-1 is a symmetric matrix
To determine if A^2n-1 is a symmetric matrix, we need to understand the properties of a symmetric matrix. A symmetric matrix A is defined as A = A^T, where A^T is the transpose of matrix A.

When we raise a skew symmetric matrix A to the power of an odd integer (2n-1), we get (A^(2n-1)) = A*A^(2n-2).

Taking the transpose of (A^(2n-1)):

((A^(2n-1))^T) = (A*A^(2n-2))^T = (A^(2n-2))^T*A^T = ((A^(2n-2))(A^T))^T

Since A is skew symmetric, A^T = -A, we can rewrite the expression as:

((A^(2n-1))^T) = ((A^(2n-2))(-A))^T = -((A^(2n-2))^T)*A^T

Since (A^(2n-2))^T = (A^(2n-2)), we have:

((A^(2n-1))^T) = -((A^(2n-2))^T)*A^T = -((A^(2n-2)))*(-A) = (A^(2n-1))

Therefore, (A^(2n-1))^T = (A^(2n-1)), which implies that A^(2n-1) is a symmetric matrix. Hence, statement b is correct.

Statement c: All positive odd integral powers of a skew symmetric matrix are skew symmetric matrix and positive even integral powers of a skew symmetric matrix are symmetric matrix.
As shown in the explanation for statement b, when we raise a skew symmetric matrix A to the power of an odd integer (2n-1), the resulting matrix (A^(2n-1)) is a symmetric matrix. Similarly, when we raise A to the power of an even integer (

Answer

Skew Symmetric Matrix:
A skew symmetric matrix is a square matrix in which the transpose of the matrix is equal to the negation of the original matrix. Mathematically, for a matrix A, it is skew symmetric if and only if A^T = -A.

Statement a) A^2n is a skew symmetric matrix:
To prove this statement, let's consider the matrix A^(2n), where n is a positive integer. We can rewrite A^(2n) as (A^n)^2.

Since A is skew symmetric, we have A^T = -A. Taking the transpose of both sides, we get (A^n)^T = (-A)^n. Now let's multiply both sides by A^n, we have A^n(A^n)^T = A^n(-A)^n.

Expanding the left-hand side, we get (A^n)(A^n)^T = A^n(-A)^n. Using the property (AB)^T = B^TA^T, we can rewrite the left-hand side as (A^n)(A^n)^T = (A^n)^TA^n.

Substituting this back into the equation, we have (A^n)^TA^n = A^n(-A)^n. Since (A^n)^TA^n = A^nA^{nT} and A^n(-A)^n = A^n(-A^n), we can simplify the equation to A^nA^{nT} = -A^nA^n.

Now, multiplying both sides by A^n, we have A^nA^{nT}A^n = -A^nA^nA^n. Simplifying this further, we get A^{nT}A^n = -A^nA^n.

Since A^{nT} = (A^n)^T and A^n is skew symmetric, we can conclude that A^{nT}A^n is also skew symmetric. Therefore, A^(2n) = (A^n)^2 is a skew symmetric matrix.

Statement b) A^2n+1 is a symmetric matrix:
To prove this statement, let's consider the matrix A^(2n+1), where n is a positive integer. We can rewrite A^(2n+1) as A^n * A^n * A.

Since A is skew symmetric, we have A^T = -A. Taking the transpose of both sides, we get (A^n)^T = (-A)^n. Now let's multiply both sides by A^n, we have A^n(A^n)^T = A^n(-A)^n.

Expanding the left-hand side, we get (A^n)(A^n)^T = A^n(-A)^n. Using the property (AB)^T = B^TA^T, we can rewrite the left-hand side as (A^n)(A^n)^T = (A^n)^TA^n.

Substituting this back into the equation, we have (A^n)^TA^n = A^n(-A)^n. Since (A^n)^TA^n = A^nA^{nT} and A^n(-A)^n = A^n(-A^n), we can simplify the equation to A^nA^{nT} = -A^nA^n.

Now, multiplying both sides by A, we have A^nA^{nT}A = -A^nA^nA

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