SOLUTION: Evaluate the following determinants. This is a 3x3 matrix. the 3 is outside of the matrix. |-2 -1 0| 3 |-3 5 -2| | 0 8 -1| Thank you.

Algebra ->  Matrices-and-determiminant -> SOLUTION: Evaluate the following determinants. This is a 3x3 matrix. the 3 is outside of the matrix. |-2 -1 0| 3 |-3 5 -2| | 0 8 -1| Thank you.      Log On


   

Question 381653: Evaluate the following determinants.
This is a 3x3 matrix. the 3 is outside of the matrix.
|-2 -1 0|
3 |-3 5 -2|
| 0 8 -1|
Thank you.
Answer by Jk22(389) About Me  (Show Source):
You can put this solution on YOUR website!
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%2C-2%2C-1%2C0%2C-3%2C5%2C-2%2C0%2C8%2C-1%29%29, we can see that a=-2, b=-1, c=0, d=-3, e=5, f=-2, g=0, h=8, and i=-1

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C-2%2C-1%2C0%2C-3%2C5%2C-2%2C0%2C8%2C-1%29%29=-22--3%2B0 Multiply

abs%28matrix%283%2C3%2C-2%2C-1%2C0%2C-3%2C5%2C-2%2C0%2C8%2C-1%29%29=-19 Combine like terms.


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Answer:

So abs%28matrix%283%2C3%2C-2%2C-1%2C0%2C-3%2C5%2C-2%2C0%2C8%2C-1%29%29=-19, which means that the determinant of the matrix %28matrix%283%2C3%2C-2%2C-1%2C0%2C-3%2C5%2C-2%2C0%2C8%2C-1%29%29 is -19


The final determinant is 3³*(-19)=the latter, hence : -513.

We can compare to the matrix multiplied by 3 :

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

--------------------------------------------------------------


From the matrix %28matrix%283%2C3%2C-6%2C-3%2C0%2C-9%2C15%2C-6%2C0%2C24%2C-3%29%29, we can see that a=-6, b=-3, c=0, d=-9, e=15, f=-6, g=0, h=24, and i=-3

Start with the general 3x3 determinant.

Plug in the given values (see above)

Multiply

Subtract

abs%28matrix%283%2C3%2C-6%2C-3%2C0%2C-9%2C15%2C-6%2C0%2C24%2C-3%29%29=-594--81%2B0 Multiply

abs%28matrix%283%2C3%2C-6%2C-3%2C0%2C-9%2C15%2C-6%2C0%2C24%2C-3%29%29=-513 Combine like terms.


======================================================================


Answer:

So abs%28matrix%283%2C3%2C-6%2C-3%2C0%2C-9%2C15%2C-6%2C0%2C24%2C-3%29%29=-513, which means that the determinant of the matrix %28matrix%283%2C3%2C-6%2C-3%2C0%2C-9%2C15%2C-6%2C0%2C24%2C-3%29%29 is -513