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Prove the following.
If A = (5 a; b 0) and A is a symmetric matrix, show that a = b.
If $A = \begin{pmatrix} 5 & a \\ b & 0 \end{pmatrix}$ and $A$ is a symmetric matrix, show that $a = b$.
To show: $a = b$
Answer
Answer
$A$ symmetric means $A^T = A$.
Hence $a = b$.
- $A^T = \begin{pmatrix} 5 & b \\ a & 0 \end{pmatrix}$, by the definition of transpose
- $A$ is symmetric, so $A^T = A$: $\begin{pmatrix} 5 & b \\ a & 0 \end{pmatrix} = \begin{pmatrix} 5 & a \\ b & 0 \end{pmatrix}$
- Equating corresponding entries (equal matrices): $b = a$
- Hence $a = b$