Find all values of a,b,c in order for A to be symmetric

Question

A= [ 4   a-b   6]
    [ 2    3    b+c ]
  [ a-b+c  5   -2]


What I dont imderstand is how you get this as an answer :

a-b=2
a-b+c=6
b+c=5

Answer 1

A square matrix, A, is symmetric when A = A^T.  So, a_ij = entry in row and column j must be the same as a_ji = entry in row j and column i.

 

a₁₂ = a₂₁  (a-b=2)

 

a₂₃ = a₃₂ (b+c=5)

 

a₃₁ = a₁₃  (a-b+c = 6)

 

Since a-b=2, a = b+2

 

Substitute for a in the equation a-b+c=6 to get b+2-b+c=6

 

                                                                             c = 4

 

Since b+c=5 and c=4, we get b = 1

 

Since a=b+2 and b=1, a = 3

 

Answer 2

A square matrix is symmetric when it is symmetric about its leading diagonal - upper left to lower right.

| x y z |

| z w j |

| k j m |

 

For A to be symmetric, 

2 = a - b

6 = a - b + c

5 = b + c