"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 :"

Question

Accepted Answer

"A^T = A, B^T = $-$B

Let A²B² $-$ B²A² = P

P^T = (A²B² $-$ B²A²)^T = (A²B²)^T $-$ (B²A²)^T

= (B²)^T (A²)^T $-$ (A²)^T (B²)^T

= B²A² $-$ A²B²

$\Rightarrow$ P is skew-symmetric matrix

$$\left[ {\matrix{ 0 & a & b \cr { - a} & 0 & c \cr { - b} & { - c} & 0 \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$

$\therefore$ 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"

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 :

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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 :

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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 :