I can find the determinant of a 2x2 and a 3x3 matrix. I can find the determinant of a 4x4 using expansion by minors, but can you find it using the diagonal method like in a 3x3? If so, how many columns do you repeat and do you proceed the same way?
Determinant of a 4x4 matrix
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masters said:
If by that statement you menat - reducing to a diagonal matrix - yes the you can do that and the method is same.
Not exactly the reply I was looking for. Let me put it this way. In a 3x3 matrix, if you repeat the 1st 2 columns, you set up a series of 3 diagonal products (northeast to southwest) - 3 diagonal products (southwest to northeast). In a 4x4, this won't work if you repeat the first 2 or the the first 3 columns and perform diagonal products and differences. My question is: Is it possible to use this method to solve for the det. of a 4x4 matrix? The matrix below has a det. of 10. I can do this my expansion by minors, but not the diagonal method. Is it possible to do it that way. If so, which columns need to be repeated?
5 2 6 3
3 9 12 1 ????
-3 1 4 1
4 1 5 3
5 2 6 3
3 9 12 1 ????
-3 1 4 1
4 1 5 3
pka
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If that mess of words means that to find the determinate of a 3x3 by expanding along the third column we ‘break’ it down onto three subdeterminates, then the answer is yes. One does a 4x4 by ‘breaking’ it down onto four subdeterminates each of which is a 3x3.
masters said:
When you evaluate the determinant of a 3 X 3 matrix using diagonals, you repeat the first two columns....be adventurous! See if repeating the first THREE columns would enable you to evaluate the determinant of a 4 X 4 matrix using diagonals.
In the BC days (before calculators) I used this method for evaluating determinants frequently.