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

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

The number of $3 \times 3$ symmetric matrices is $7^{6}$

The number of $3 \times 3$ matrices is $7^{9}$

The number of $3 \times 3$ skew symmetric matrices is $7^{3}$

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

answer is A, B, C, D.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

trace is ‘0’ if diagonala elements are (0,0,0), (-1,0,1),(-2,2,2),(-3,0,3),(1,1,-2),(-1,-1,2) in any order.
No. of ways = $(1 + 5 \times 3 + 2 \times \frac{3}{2}) = 37$
Each of remaining 6 elements can be taken in 7 ways No. of ways = $7^{6} \Rightarrow$ Required = $37 \times 7^{6}$ .
B) No. of $3 \times 3$ matrices is $7^{9}$ (each element can be taken 7 ways).
C) In a skew-synmetric matrix each diagonal elements must be 0. The each of the 3 elements above the principal diagonal can be taken in 7 ways, and correspondingly elements below the principal diagonal can be taken in only 1 way.
Required = $1 \times 7^{3} \times 1 = 7^{3}$ $(a_{i j} = - a_{j i} \forall i, j)$
D) In a symmetric matrix, each of the diagonal element can be taken in 7 ways also each of the elements above the principle diagonal can be taken in 7 ways and correspondingly elements below the principle diagonal can be taken in only 1 way.
Required= $7^{3} \times 7^{3} \times 1 = 7^{6}$ .