(D) Given that $A$ is a skew-symmetric matrix,we have $A^T = -A$. For $A^{2n}$: $(A^{2n})^T = (A^T)^{2n} = (-A)^{2n} = (-1)^{2n} A^{2n} = A^{2n}$. Sin

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(D) Given that $A$ is a skew-symmetric matrix,we have $A^T = -A$. For $A^{2n}$: $(A^{2n})^T = (A^T)^{2n} = (-A)^{2n} = (-1)^{2n} A^{2n} = A^{2n}$. Since $(A^{2n})^T = A^{2n}$,$A^{2n}$ is a symmetric matrix. Thus,statement $1$ is false. For $A^{2n+1}$: $(A^{2n+1})^T = (A^T)^{2n+1} = (-A)^{2n+1} = (-1)^{2n+1} A^{2n+1} = -A^{2n+1}$. Since $(A^{2n+1})^T = -A^{2n+1}$,$A^{2n+1}$ is a skew-symmetric matrix. Thus,statement $2$ is true. Therefore,option $D$ is correct.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Special types of matrices, Transpose of matrices and Trace of matrices

Medium

If $A$ is a skew-symmetric matrix,then (given $n \in N$):
$1$. $A^{2n}$ is a skew-symmetric matrix.
$2$. $A^{2n+1}$ is a skew-symmetric matrix.

AP EAMCET 2020 All AP EAMCET

A
$1$ is true,$2$ is false
B
Both $1$ and $2$ are true
C
Both $1$ and $2$ are false
D
$1$ is false,$2$ is true

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3 and 4 .Determinants and Matrices

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Special types of matrices, Transpose of matrices and Trace of matrices

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If $A$ is a skew-symmetric matrix,then (given $n \in N$): $1$. $A^{2n}$ is a skew-symmetric matrix. $2$. $A^{2n+1}$ is a skew-symmetric matrix.

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Express the matrix $A = \left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.

If $A$ and $B$ are square matrices of the same order,then which of the following properties holds true for the transpose of their product?

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If $A$ is a skew-symmetric matrix,then (given $n \in N$):
$1$. $A^{2n}$ is a skew-symmetric matrix.
$2$. $A^{2n+1}$ is a skew-symmetric matrix.