## find values for a, b, and c for which matrix A is symmetric

Linear spaces and subspaces, linear transformations, bases, etc.
testing
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### find values for a, b, and c for which matrix A is symmetric

find values for a, b, and c for which the matrix A is symmetric
matrix A =   2   a-2b+c   2a+b+c
3    5        a+c
0   -2         7
i know that a-2b+c=3 and son on ... but i cant seem to process the equation to get answers consistent for the equation.

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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For the matrix to be symmetric, it must be equal to its own transpose. In other words:
[ 2  a-2b+c  2a+b+c ]   [    2     3   0 ]
[ 3     5     a+c   ] = [ a-2b+c   5  -2 ]
[ 0    -2      7    ]   [ 2a+b+c  a+c  7 ]
By equating entries (which is the definition of "matrix equality"), we get the following system:
1a - 2b + 1c =  3
2a + 1b + 1c =  0
1a + 0b + 1c = -2
Solve the linear system for the values of "a", "b", and "c".
1a - 2b + 1c =  3
2a + 1b + 1c =  0
1a + 0b + 1c = -2

-R3 + R1 -> R1

- 2b      =  5
2a + 1b + 1c =  0
1a + 0b + 1c = -2

1b      =  -2.5
2a + 1b + 1c =   0
1a +    + 1c =  -2

-R1 + R2 -> R2

1b      =  -2.5
2a + 0b + 1c =   2.5
1a +    + 1c =  -2

-(1/2)R2 + R3 -> R3

1b         =  -2.5
2a      +    1c =   2.5
-(1/2)c =  -3.25
...and so forth.

Then plug these values back into the original expression of the matrix A.