Let A and B be 3 [] 3 real matrices such that A is symmetric matrix and B is skew-symmetri…

Question

Let A and B be 3 $$\times$$ 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 - B2A2) X = O, where X is a 3 $$\times$$ 1 column matrix of unknown variables and O is a 3 $$\times$$ 1 null matrix, has :

Accepted Answer

AT = A, BT = -B

Let A2B2 - B2A2 = P

PT = (A2B2 - B2A2)T = (A2B2)T - (B2A2)T

= (B2)T (A2)T - (A2)T (B2)T

= B2A2 - A2B2

$$ \Rightarrow $$ P is skew-symmetric matrix

$$\left[ {\begin{pmatrix} 0 & a & b  \
   { - a} & 0 & c  \
   { - b} & { - c} & 0  \

\end{pmatrix} } \right]\left[ {\begin{pmatrix} x  \
   y  \
   z  \

\end{pmatrix} } \right] = \left[ {\begin{pmatrix} 0  \
   0  \
   0  \

\end{pmatrix} } \right]$$

$$ \therefore $$ ay + bz = 0 ..... (1)

-ax + cz = 0 .... (2)

-bx -cy = 0 ..... (3)

From equation 1, 2, 3

$$\Delta$$ = 0 \& $$\Delta$$1 = $$\Delta$$2 = $$\Delta$$3 = 0

$$ \therefore $$ equation have infinite number of solution

Solutions · 1