In Problems 7-14, find the value of each determinant.

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Key Concepts

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix, such as whether it is invertible. It is an essential concept in linear algebra with applications in solving systems of equations, understanding matrix transformations, and evaluating eigenvalues.

3x3 Determinant Calculation

Calculating the determinant of a 3x3 matrix generally involves using the rule of Sarrus or cofactor expansion. This process involves multiplying combinations of elements from the matrix and performing additions and subtractions in a specific pattern. Mastery of 3x3 determinant calculations is crucial as it serves as a foundation for understanding determinants of higher order matrices.

Cofactor Expansion

The cofactor expansion, also known as Laplace's expansion, is a method for computing determinants. It involves selecting a row or column and summing the products of its elements with their corresponding cofactors, which are determinants of smaller submatrices. This technique is particularly useful for understanding the recursive nature of determinants and for matrices larger than 3x3.

Transcript

0:00 Hello.

00:02 So here we are finding the determinant of a three by three matrix.

00:06 So what we're going to do is we're going to expand along the first row.

00:10 So we take three and then times the determinant of its minor.

00:15 We cross out the row and the column where three is in and then look at the remaining two by two matrix.

00:22 So therefore we have three again times the determinant of its minor.

00:28 So that's going to be negative one, five, 2, negative 2.

00:32 So negative 1, 5, 2, negative 2.

00:38 Okay.

00:38 And then we have minus the second entry in the in the row, which is 4.

00:45 So 4 is positive.

00:47 So it's minus 4, minus 4 times the determinant of its minor.

00:53 So we cross out again, the first row, second column, and look at its corresponding 2x2 matrix, which will be 1 -5 -1 -2.

01:02 So 1 -5 -1 -2.

01:05 And then plus the last member in the row, which is 2 times the determinant of its minor, which is 1 -1 -1 -2.

01:17 Okay, so now what do we have? we have three times the determinant of, well, a 2 -by -2 matrix is just the product of the main diagonal minus the product of the other diagonal.

01:28 So we have negative 1 times negative 2, which is a positive 2, and then minus 5 times 10.

01:35 So that's going to be 3 times times 2 and then minus 5 times 2...

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