Matrices · Question 18

Let M denote the set of all real matrices of order 3 3 and let S = \-3, -2, -1, 1, 2\. Let S1 = \A = [aij] M : A = AT and aij S, i, j\, S2 = \A = [aij] M : A = -AT and aij S, i, j\, S3 = \A = [aij] M : a11 + a22 + a33 = 0 and aij S, i, j\. If n(S1 S2 S3) = 125 , then equals .

Answer

1613

Solution

S2: a skew-symmetric matrix has aii = 0, but 0 S S2 = .

S1: a symmetric matrix is fixed by its 3 diagonal and 3 upper entries n(S1) = 56.

S3: the 6 off-diagonal entries are free 56 ways.

Diagonal triples from S with sum 0:

\-3, 1, 2\: 3! = 6; \-2, 1, 1\: 3; \-1, -1, 2\: 3 12 ways

n(S3) = 12 56

S1 S3: symmetric with trace 0 n(S1 S3) = 12 53

n(S1 S2 S3) = 56 + 12 56 - 12 53

= 53(125 + 1500 - 12) = 125 1613

= 1613

Answer: 1613