Find the determinants of the matrices A. [ 0 0 0 0 1 0 0 2

Find the determinants of the matrices A. [ 0 0 0...

Key Concepts

Determinant

A determinant is a scalar value that is computed from the elements of a square matrix. It encapsulates important properties of the matrix, such as whether the matrix is invertible, and it provides a scaling factor for the linear transformation represented by the matrix. The determinant is calculated as a weighted sum over all permutations of the matrix entries, where each term is multiplied by a sign factor depending on the parity of the permutation.

Permutation Sign in Determinants

In the determinant formula, each term in the sum corresponds to a permutation of the row or column indices. The sign assigned to each permutation, which is positive for even permutations and negative for odd ones, plays a crucial role in determining the correct value of the determinant. This concept is pivotal because it ensures that the contributions from different arrangements of matrix entries are combined accurately.

Sparse Matrix Evaluation

Sparse matrices, which contain a large number of zero elements, can simplify the calculation of determinants. Because many of the terms in the determinant expansion will be zero, identifying the nonzero structural pattern in the matrix allows for a more efficient evaluation. This often involves recognizing which permutations yield nonzero products and focusing the computation only on those terms.

Transcript

00:01 In the given question, we are told to find the determinant of the matrix a that is given over here.

00:08 The matrix is a 5 by 5 matrix and it has the elements 0 0 0 0 1, 0 0 2 0 1, 2 0 0, 0 4 0 0 0, then you have 0 0 0 3 0 and finally you have 5 0 0 0 0.

00:52 Now to find the determinant of this matrix, what you do is, let's expand this along the first row, right.

01:03 So when you take the first row, you can see that all the elements except this last element is non -zero.

01:11 So you can just take the expansion with respect to this element since all the elements will just give us 0 itself on multiplying.

01:20 So the way we expand this determinant is, you would take, let's say, what you do is, let's just write this as the determinant of a, let's remove this matrix sign and now what you do is, you take 1 and you take the minor of 1 and multiply it with 1, right.

01:54 So to find the minor of 1, what you do is, you delete the row that 1 is in, you delete the column that 1 is in and now the determinant of the remaining elements is what you call the minor of 1.

02:09 So over here, what you would have then is 1 times the minor of 1 will have the elements 0 0 2 0, 0 4 0 0, 0 0 0 3 and 5 0 0 0.

02:32 So again, we have now a 4 by 4 matrix.

02:36 So now, we are again expanding this with the first row.

02:44 One thing that you should notice is that we assign different signs to the different elements in a matrix when we expand.

02:51 So we assign plus to the first element, then we alternate the sign with each element.

02:57 So the second element is taken as minus, the third element is taken as plus, the fourth element is taken as minus, the fifth element will be taken as plus.

03:09 So these signs are just used or assigned when we expand this determinant, right.

03:16 So then in the case of this row, you have plus, minus, plus over here...

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