Question

if \\(a = a^t\\) and \\(b = b^t\\), which one of the following matrices is certainly symmetric?

• \\(ab\\)
• \\(ba\\)
• \\(abab\\)
• \\(aba\\)

Explanation:

Define symmetric matrix condition

Using the Matrix Classification knowledge point, a matrix \(M\) is symmetric if and only if:

$$ M^T = M $$

We are given that \(A^T = A\) and \(B^T = B\).

Analyze transpose of AB

We apply the transpose product rule:

$$ (AB)^T = B^T A^T $$

Substituting the given symmetric properties:

$$ (AB)^T = BA $$

Since matrix multiplication is not generally commutative (\(BA
eq AB\)), \(AB\) is not certainly symmetric.

Analyze transpose of BA

Similarly, for the product \(BA\):

$$ (BA)^T = A^T B^T = AB $$

Since \(AB
eq BA\) in general, \(BA\) is not certainly symmetric.

Analyze transpose of ABAB

For the product \(ABAB\):

$$ (ABAB)^T = B^T A^T B^T A^T = BABA $$

Since \(BABA
eq ABAB\) in general, \(ABAB\) is not certainly symmetric.

Analyze transpose of ABA

For the product \(ABA\):

$$ (ABA)^T = A^T B^T A^T $$

Substituting \(A^T = A\) and \(B^T = B\):

$$ (ABA)^T = ABA $$

Since \((ABA)^T = ABA\), the matrix \(ABA\) is certainly symmetric.

Answer:

• (A) AB
• (B) BA
• (C) ABAB
• (D) ABA (Correct answer)