In 14-18, assume the entries of all matrices are real numbers.

Prove that if A is the m × m symmetric matrix, then A²is symmetric.

Prove that if A Is an m × m symmetric matrix, then A² Is symmetric.

Given information:

Assume the entries of all matrices are real numbers.

Proof:

Given: A Is an m x m symmetric matrix

To prove: A² Is symmetric

PROOF:

Since A=(a ij) is an m×m symmetric matrix: aij=aji for all i,j=1,2,...,mLet us then determine if the element of A2=(b ij) also symmetric property (b ij=b ji).

bij=ijth entry of A2    =ijth entry of A⋅A    =∑r=1na ira rj    =∑r=1n