# Proof of symmetric and anti symmetric matrices

1. Aug 31, 2011

### prawinath

1. The problem statement, all variables and given/known data

aij is a symmetric matrix
bij is a an anti symmetric matrix

prove that aij * bij = 0

2. Relevant equations

aij * bij

3. The attempt at a solution

any one got any ideas ?

2. Aug 31, 2011

### Hootenanny

Staff Emeritus
HINT: What happens when you interchange the indices?

3. Aug 31, 2011

### Fredrik

Staff Emeritus
$a_{ij}$ doesn't denote a matrix. It denotes the component on row i, column j, of a matrix.

Since $$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\neq 0,$$ it's not true that the product of a symmetric and an antisymmetric matrix is =0. On the other hand, it is true that $a_{ij}b_{ij}=0$ (assuming that repeated indices are summed over). You should take some time to think about what the expression $a_{ij}b_{ij}$ really means, and what matrix operation(s) it involves.

Do you know the definition of matrix multiplication? If $a_{ij}$ denotes a component of a matrix A, and $b_{ij}$ denotes a component of a matrix B. Then what will you find on row i, column j of AB?