Find the determinant of the matrix. Expand by cofactors on...

Find the determinant of the matrix. Expand by cofactors on each indicated row or column.
$\left[\begin{array}{rrrr}10 & 8 & 3 & -7 \\ 4 & 0 & 5 & -6 \\ 0 & 3 & 2 & 7 \\ 1 & 0 & -3 & 2\end{array}\right]$
(a) Row 3
(b) Column I

Key Concepts

Determinant of a Matrix

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible, the scaling factor for linear transformations, and properties related to eigenvalues. It plays a significant role in solving systems of linear equations, changing variables in integrals, and understanding geometric properties of transformations.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method for calculating the determinant of a matrix by expanding it along a row or column. This technique involves multiplying each element of the chosen row or column by its corresponding cofactor, which accounts for its position in the matrix, and summing these products to obtain the determinant.

Minor of a Matrix

In the context of determinants and cofactors, a minor of an element is the determinant of the smaller matrix that results when the row and column of that element are removed. The minor serves as a building block in calculating the cofactor, which is the minor adjusted by a sign factor based on the element’s position in the matrix.

Sign Pattern in Cofactor Expansion

The sign pattern in cofactor expansion is determined by the checkerboard arrangement of plus and minus signs starting with a positive sign in the top-left corner. Each cofactor is computed by multiplying the minor of an element by either +1 or -1, depending on its row and column indices, which is critical in ensuring that the overall determinant is calculated correctly.

Transcript

00:03 So, in order to expand this matrix along the row 3, row 3 is i can see this one.

00:11 So i will be writing first 0 into something that is 0 itself, then this is plus minus plus, this will be minus.

00:20 So minus 3 times, this will come as 10, 4, 1.

00:29 Next this column will be blocked.

00:32 Then i have 3, 5, minus 3, minus 7, minus 6 and 2.

00:43 In the last, then plus 2, this is 10, 4, 1 as it is.

00:56 Then i have 8, 0 and the last entry will be minus 7, minus 6 and 2.

01:05 And the last entry will be with 7 that will come with a negative sign minus 7 10 4 1 800 and lastly it will be 3 5 minus 3 so this value on simplification will be simply equal to 0 first of all minus 3 times this determinant once solved will give me so this determinant value on solving will come out to be minus 3 this determinant can be solved as 10 plus minus.

01:48 So this will be minus 8.

01:50 I'll let 4 into 2, that is 8 plus 6.

01:58 And finally, it will be minus 7 into, this will be minus 8, multiplied by 4 into 3, that is minus 12, minus 5.

02:13 So, this gets me the negative sign and finally on simplification i will have it as 9.

02:21 This values will be, so this will be 9 and this will be minus 224 and this is minus 952.

02:32 Simple calculations, this will yield me minus 1167.

02:36 So that is my answer...