Evaluate each determinant by expanding by cofactors.

Key Concepts

Minor

A minor of an element in a matrix is the determinant of the submatrix that remains after removing the element's row and column. Minors are used in the process of calculating cofactors, which play a crucial role in the cofactor expansion method.

Cofactor

A cofactor is obtained by multiplying the minor of an element by a sign factor, which is determined by the element’s position using the formula (-1)^(i+j), where i and j are the row and column indices. Cofactors adjust the sign of the minors appropriately when they are used in the cofactor expansion of a determinant.

Determinant

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether it is invertible and by measuring the scaling factor of the linear transformation described by the matrix.

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method for evaluating the determinant of a matrix. It involves breaking down the determinant into smaller determinants (minors) multiplied by their corresponding cofactors, and then summing these products. This method is particularly useful for matrices of size 3x3 or greater.

Transcript

00:03 So here we have this 3x3 square matrix and we are asked to find the determinant of it.

00:08 We would find the determinant of this matrix by using the method expanding by cofactors.

00:15 So expanding by cofactors tells us that the determinant is a11 -1 -c -1 plus a12, c -12, plus a -13, c -13, if you're not sure what all this means, you'll figure it out as we go along.

00:37 So first, a11 means element in row 1, column 1, which is simply the number 1.

00:45 So we have 1.

00:48 And then c11, because 1 plus 1 or the i and j equals 2, it's an even number.

00:55 So c11 is the same thing as minor 11.

00:59 M1 can be found by deleting row 1.

01:04 And column 1.

01:07 So if we delete row 1 in column 1, we are simply left with 5, negative 2, 3, negative 2.

01:20 Now, a1, is element in row 1, column 2, which is 3, and c1, because 1 and 2, 1 plus 2 equals 3, which is an odd number, c12 is a negative version of m1.

01:39 So we have negative m12.

01:43 M12 is found again by deleting row 1, but this time by deleting column 2.

01:50 So we have negative 2, 3, 3, negative 2.

01:56 Lastly, we have a13 or element in row 1, column 3, which is 1.

02:05 Times c13 because 1 plus 3 equals 4, which is an even number, we are simply finding m13.

02:12 So m13 is found by deleting the first row and the third column.

02:18 So we have negative 2, 3, 5, negative 2.

02:25 Now we have to find the determinants of all these cofactors.

02:30 So we have, oh, sorry, scrolling issues...