Now, I understand algebraic operations within matrices and determinants of matrices, but for some reason I can't seem to get this determinant right through making it a triangular matrix and then multiplying the elements along the diagonal.
First off, I start with:
| 1 3 9 |
| 1 -1 8|
|1 4 4 |
The determinant is apparently 21, but when I make it a triangular matrix I get 84.
My work for making it triangular:
| 1 3 9 |
| 1 -1 8|
|1 4 4 |
becomes
| 1 3 9 |
| 0 -4 -1|
|0 1 -5 |
I subtracted the 1st row from the 2nd for my new 2nd, and the 1st row from the 3rd for my new 3rd.
Next:
| 1 3 9 |
| 0 -4 -1|
|0 0 -21 |
I multiplied the 3rd row by 4, added it to the 2nd row and declared that as my new 3rd row. Now, the determinant of my triangular matrix is (1)(-4)(-21) = 84 which is not 21.
What am I doing wrong?
Answers & Comments
Verified answer
Your last step is not a valid row operation.
If you multiply 4 times row 3 then add it to 2, you get a new row 2 not a new row 3.
1 3 9
0 0 -21
0 1 -5
Then you exchange row 3 with row 2. A row exchange causes the determinant to be the negative of what you had before.
1 3 9
0 1 -5
0 0 -21
Multiplying the diagonals gives you -21 which is the negative of the determinant of the original matrix.
Alternatively, you could have taken 1/4 of row 2 added to row 3.
1 3 9
0 -4 -1
0 0 -21/4
The diagonals multiply to give 21
well im not sure how to do it that way but its a 3x3 matrix so the determinant is:
you take a number (changing the sign of the number of the even columns) of the top row...cover up the row and column that it is in and multiply that number by the determinant of the 2x2 matrix that is left....add the three numbers together to get the determinant
1*|-1 8| + 3 |1 8| + 9 |1 -1|
...|4 4|.........|1 4|.........|1 4|
1(-36) + (-3)(-4) + 9(5)
-36 + 12 + 45
21