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If a matrix A is both symmetric and skew symmetric, then

If a matrix A is both symmetric and skew symmetric, then Correct Answer A is a zero matrix

Concept:

Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself

⇒ AT = A 

Square matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A

⇒ AT = −A

Calculation:

Given: Matrix A is both symmetric and skew-symmetric.

A is a symmetric matrix if A=AT

A square matrix A is said to be symmetric if aij = aji for all i and j

Where aij and aji is an element present in the matrix.

For the skew-symmetric matrix,

AT = −A

Similarly, A square matrix A is said to be symmetric if aij = -aji for all i and j

Where aij and aji is an element present in the matrix.

From above we can see that,

aij= -aij

⇒ 2aij=0

⇒ aij=0.

∴ Matrix A is the zero matrix or Null matrix.

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