Question

If Aij is a square matrix such that aij = i2 - j2 then write whether A is a symmetric or skew symmetric.

Answer 1

Answer:

A is not a symmetric matrix but it is skew symmetric

Step-by-step explanation:

Concept:

A square matrix A is saaid to be symmetric if $A=A^T$

A square matrix A is saaid to be skew symmetric if $A=-A^T$

$Given:\\\\a_{ij}=i^2-j^2\\\\a_{11}=1^2-1^2=1-1=0\\\\a_{12}=1^2-2^2=1-4=-3\\\\a_{13}=1^2-3^2=1-9=-8\\\\a_{21}=2^2-1^2=4-1=3\\\\a_{22}=2^2-2^2=4-4=0\\\\a_{23}=2^2-3^2=4-9=-5\\\\a_{31}=3^2-1^2=9-1=8\\\\a_{32}=3^2-2^2=9-4=5\\\\a_{33}=3^2-3^2=9-9=0$

$Now\\\\A=\left[\begin{array}{ccc}0&-3&-8\\3&0&-5\\8&5&0\end{array}\right]........(1)\\\\\\A^T=\left[\begin{array}{ccc}0&3&8\\-3&0&5\\-8&-5&0\end{array}\right].......(2)\\\\ \\-A^T=\left[\begin{array}{ccc}0&-3&-8\\3&0&-5\\8&5&0\end{array}\right]........(3)$

From (1) and (2)

$A\neq A^T$

Therefore A is not a symmetric matrix

From (1) and (3)

$A=-A^T$

Therefore A is a skew symmetric matrix