Q.
Let A and B be 3×3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations A2B2−B2A2X=0, where X is a 3×1 column matrix of unknown variables and O is a 3×1 null matrix, has
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a
a unique solution
b
infinitely many solutions
c
no solution
d
exactly two solutions
answer is B.
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Detailed Solution
Given AT=A, BT=-B given system is homogenous system of equationsDeterminant of coefficient matrix is A2B2−B2A2Consider A2B2−B2A2T=A2B2)T−(B2A2T =B2A2-A2B2 =-A2B2−B2A2Therefore, A2B2−B2A2 is a skew symmetric matrix of odd orderHence the determinant of the coefficient matrix is zero. Therefore, the system of equations have non trivial solutions, and it has infinitely many solutions
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