Evaluate the determinant of the given matrix by first using...

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form.
$$\begin{array}{|lllll|}
1 & 2 & 3 & 4 & 5 \\
5 & 4 & 3 & 2 & 1 \\
2 & 3 & 4 & 5 & 6 \\
6 & 5 & 4 & 3 & 2 \\
0 & 2 & 4 & 6 & 8
\end{array} |$$

Key Concepts

Properties of Determinants

The properties of determinants include their behavior under row operations, their multiplicative nature with respect to matrix multiplication, and the rule that swapping two rows results in a negation of the determinant. Understanding these properties is essential when manipulating matrices to simplify determinant calculations, ensuring that any changes to the determinant's value are properly accounted for.

Upper Triangular Form

An upper triangular matrix is a square matrix in which all the elements below the main diagonal are zero. Transforming a matrix to this form using elementary row operations makes it easier to compute the determinant, as the determinant of an upper triangular matrix is simply the product of the diagonal elements.

Elementary Row Operations

Elementary row operations are the basic operations that can be applied to rows of a matrix. They include swapping two rows, multiplying a row by a nonzero scalar, and adding a scalar multiple of one row to another row. These operations are used to simplify matrices and solve linear systems, and while they can alter the determinant, certain adjustments can be made to account for these changes.

Determinant

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible, the scaling factor of the linear transformation represented by the matrix, and in some cases the volume change when the matrix is seen as a transformation in space.

Transcript

00:01 Section 3 .2, problem number 14, we're asked to find the determinant of a 5x5 matrix using elementary row operations to get it to upper triangular form and then find the determinant just by looking along the diagonal.

00:16 So what i want to do here is just i like to go and just clear the first row.

00:21 So everything under this 1 i need to turn into a 0.

00:25 So the row operations will do that for me would be minus 5 times row 1.

00:32 Plus row 2 and that will become the new row 2.

00:39 So that will clear out this 5 for me.

00:42 So let's do that operation.

00:45 We're going to have 1, 2, 3, 4, 5.

00:50 So negative 5 plus 5 is 0.

00:54 Negative 5 times 2 is negative 10.

00:57 Negative 10 plus 4 is negative 6.

01:02 And then negative 5 times 3 is negative 15.

01:07 15 plus 3 is negative 12 and then negative 5 times 4 is negative 20 negative 20 plus 2 negative 18 and then negative 5 times 5 is negative 25 negative 25 plus 1 negative 24 now to clear out this element here which is a 2 you could do negative 2 times row 1 plus row 3 and that will become our new row 3 so this gives me a 0 0 0 0 0 0 0.

01:44 Negative 2 times 2 is negative 4, and negative 4 plus 3 is negative 1.

01:51 Negative 2 times 3 is negative 6, negative 6 plus 4 is negative 2.

02:02 Negative 2 times 4 is negative 8.

02:05 Negative 8 plus 5 is negative 3.

02:09 And then negative 2 times 5, negative 10, negative 10 plus 1 is, let's say negative 2...

02:17 Negative 10, negative 10 plus 6 is negative 4.

02:28 Now to clear out the 6, you're going to have negative 6 row 1 plus row 4...

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