SOLUTION: Can you help me with this one? Show that in any 3x3 determinant if one row is 0, the value of the determinant is 0.

Algebra ->  Matrices-and-determiminant -> SOLUTION: Can you help me with this one? Show that in any 3x3 determinant if one row is 0, the value of the determinant is 0.      Log On


   

Question 888768: Can you help me with this one?
Show that in any 3x3 determinant if one row is 0, the value of the determinant is 0.
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
0,0,0
1,2,3
4,5,6
Solved by pluggable solver: Finding the Determinant of a 3x3 Matrix

If you have the general 3x3 matrix:

%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29

the determinant is:

Which further breaks down to:



Note: abs%28matrix%282%2C2%2Ce%2Cf%2Ch%2Ci%29%29, abs%28matrix%282%2C2%2Cd%2Cf%2Cg%2Ci%29%29 and abs%28matrix%282%2C2%2Cd%2Ce%2Cg%2Ch%29%29 are determinants themselves.
If you need help finding the determinant of 2x2 matrices (which is required to find the determinant of 3x3 matrices), check out this solver

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From the matrix %28matrix%283%2C3%2C0%2C0%2C0%2C1%2C2%2C3%2C4%2C5%2C6%29%29, we can see that a=0, b=0, c=0, d=1, e=2, f=3, g=4, h=5, and i=6

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C0%2C0%2C0%2C1%2C2%2C3%2C4%2C5%2C6%29%29=0-0%2B0 Multiply

abs%28matrix%283%2C3%2C0%2C0%2C0%2C1%2C2%2C3%2C4%2C5%2C6%29%29=0 Combine like terms.


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Answer:

So abs%28matrix%283%2C3%2C0%2C0%2C0%2C1%2C2%2C3%2C4%2C5%2C6%29%29=0, which means that the determinant of the matrix %28matrix%283%2C3%2C0%2C0%2C0%2C1%2C2%2C3%2C4%2C5%2C6%29%29 is 0