Find all values of the unknown constants a, b, and c in order for the following

Question

Find all values of the unknown constants a, b, and c in order for the following 3 x 3 matrix A to be symmetric:

A= [2 a-2b+2c 2a+b+c]
|3 5 a+c |
[0 -2 7 ]

Explanation / Answer

A = [2, a-2b+2c, 2a+b+c] ....|3, 5, a+c | ....[0, -2, 7 ] For A to be symetric you need to have: (1) a-2b+2c = 3 (2) 2a+b+c = 0 (3) a+c = -2 Easiest is to use the 3rd equation a+c =-2 to start as it means c=-a-2= -(a+2) and you can carry that into the 2nd equation and get: (2) 2a + b + (-a-2) = 0 => 2a+b-a-2=0 => a + b = 2 => b = 2-a Next carry the results (b = 2-a) and (c = -a-2) into the first equation (1) a + -2(2-a) +2(-a-2) = 3 => a -4 + 2a -2a - 4 = 3 => a - 8 = 3 => a=11 Taking this back to (b = 2-a) gives: b = -9 And going back to (3) you get: c = -13 VERIFY: (1) 11-2(-9)+2(-13) = 11+18-26 = 29-26 = 3 OK (2) 2x11 +(-9) + (-13) = 22 -9 -13 = 22-22 = 0 OK (3) 11 + (-13) = 11-13 = -2 OK A = [2, 3, 0] ....|3, 5, -2 | ....[0, -2, 7 ]

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