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

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

Postby testing on Fri Mar 20, 2009 1:26 pm

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.
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Postby stapel_eliz on Fri Mar 20, 2009 4:25 pm

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.

:D
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