Using the elements –3, –2, –1 0, 1, 2, 3

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The number of 3 × 3 skew symmetric matrices is 7³

The number of 3 × 3 matrices having trace 0 is $37 (7^{6})$

The number of 3 × 3 matrices is 7⁹

The number of 3× 3 symmetirc matrices is 7⁶

answer is A, B, C, D.

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Detailed Solution

Using the 7 elements total number of 3 by 3 matrices formed=7⁹

For symmetric matrices each place in the Upper(lower) triangle and diagonal can be filled in 7 ways. So number of symmetric matrices are 7⁶ ways

For skew symmetric matrices elements in principal diagonal are filled with 0 in one way. The elements in principal diagonal can be filled with 7 elements in 7³ ways.

The trace is 0.

The elements in principal diagonal can be filled using the following

elements in principal diagonal such that trace is zero	Number of possibilities
(0,0,0)	1
(-1,0,1),(-2,0,2),(-3,0,3)	3(3!)=18
(1,1,-2), (-1,-1,2)	2(3)=6
(1,2,-3)(-1,-2,3)	2(6)
Total	37

Each of the non diagonal element can be filled by using the 7 elements in 7⁶ ways.

The number of 3 × 3 matrices having trace 0 is $37 (7^{6})$

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