Question

If A is a symmetric matrix and $n\in N$ then ${{A}^{n}}$ is
A. Symmetric
B. Skew symmetric
C. Diagonal matrix
D. None of these

Answer 1

Hint: Here we check whether the given symmetric matrix is satisfying the property of the symmetric matrix. A symmetric matrix is a square matrix equal to its transpose.

Formula Used:
If A is Symmetric Matrix then it means $A=A^T$

Complete step-by-step solution:
The given matrix is symmetric, therefore
$A=A^T$
Thus,
${{({{A}^{n}})}^{T}}={{({{A}^{T}})}^{n}}={{(A)}^{n}}$
Therefore, ${{A}^{n}}$
is also a symmetric matrix.
Hence, for all $n\in N$, $A^n$ is also symmetric.

So, option is A correct.

Additional Information:
Symmetric Matrix Properties
1. These are the key characteristics of symmetric matrices that set them apart from other kinds of matrices. These properties are listed as,
2. A symmetric matrix is obtained as the sum/difference matrix of two symmetric matrices.
3. When two symmetric matrices $A$ and $B$ are multiplied, the product matrix $AB$ is symmetric if and only if the two matrices are commutative, that is, if$ AB = BA$.
4. For any integer $n$, if $A$ is symmetric, $A^n$ is also symmetric.
5. If a matrix $A$'s inverse exists, it will only be symmetric if the square matrix $A$ is symmetric.

Note: For any integer $n$, if $A$ is symmetric, $A^n$ is also symmetric. If and only if a matrix equals its transpose, it is said to be symmetric. In a symmetric matrix, all entries above the main diagonal are reflected into identical entries below the diagonal. Only when a matrix is the opposite of its transpose is it skew-symmetric. In a skew-symmetric matrix, the main diagonal entries are all zero.