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smize
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I am wondering how one would find a the determinant of a 4x4 or greater. This isn't an urgent question, just a curiosity.
voko said:
AlephZero said:A much more efficient way is to do row operations on the matrix which don't change the value of the determinant (or only multiply it by -1), but systematically change the matrix so that all the entries below the diagonal are zero. The determinant is then just the product of the diagonal terms. In the worst case, that takes about n3 operations. For a 10 x 10 matrix, n3 = 1,000 and n! = about 3.6 million, so one way is about 3600 times faster than the other!
AlephZero said:That is "correct", and an interesting theoretical result, but it's a hopelessly inefficient way to calculate the determinant of a general matrix, because it takes of the order of n! operations for an n x n matrix.
voko said:Whenever I get to compute a det with n >= 3 MANUALLY, I use this method. Possibly because I remember it by heart. Doing arbitrary n dets most efficiently is a distinctly different matter.
There are several methods for finding the determinant of a 3x3 matrix, including the Laplace expansion method, the diagonal method, and the row reduction method. These methods involve different operations on the elements of the matrix, such as multiplying and adding rows or columns.
Yes, the determinant of a 3x3 matrix can be negative. The sign of the determinant depends on the order in which the elements are arranged in the matrix. If the elements are arranged in a clockwise order, the determinant will be positive, and if they are arranged in a counterclockwise order, the determinant will be negative.
The determinant of a matrix is closely related to its invertibility. A matrix is invertible if and only if its determinant is non-zero. This means that if the determinant of a matrix is zero, the matrix is not invertible, and it is said to be singular.
Changing the elements of a matrix can have a significant effect on its determinant. Multiplying a row or column by a constant will result in the determinant being multiplied by the same constant. Swapping two rows or columns will change the sign of the determinant. Adding a multiple of one row or column to another will not change the determinant.
Yes, there are a few shortcuts for finding the determinant of a 3x3 matrix. One method is to use the diagonal method, where you multiply the elements along both diagonals and then take the difference of the two products. Another method is to use the rule of Sarrus, where you write the first two columns of the matrix next to each other and then add the products of the diagonals going down and subtract the products of the diagonals going up.