Evaluate the determinant of the given matrix by first using elementary row operations to reduce

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

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the entries below the main diagonal are zero. Its determinant is simply the product of the entries on the main diagonal, which simplifies the computation significantly. Reducing a matrix to an upper triangular form using elementary row operations provides a straightforward way to evaluate its determinant.

Elementary Row Operations

Elementary row operations—swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another—are used to simplify matrices without changing (or by predictably altering) their determinants. These operations are crucial in methods like Gaussian elimination and help in reducing a matrix to a form where the determinant is easier to compute.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and encapsulates key properties such as invertibility and volume scaling. It is fundamental in linear algebra and can be used to determine if a system of linear equations has a unique solution, as well as to understand how linear transformations affect space.

Transcript

00:01 Section 3 .2, property of determinants, problem number six, we're given a 3x3 matrix and we're asked to use elementary row operations to reduce it to upper triangular form so that the determinant is just the product of all the numbers that are on the diagonal.

00:16 So first step is i need to start clearing, so let's clear this number right here.

00:22 So in order to do that, if i took what negative five -thirds times row one plus row two and place that answer back into row two, that would give me the desired effect there.

00:40 So that would give me three, seven, one.

00:43 Negative five -thirds times three is negative five plus negative five is zero.

00:47 That was by design.

00:49 Negative five -thirds times seven.

00:51 So that's going to be what negative 35 thirds so negative 35 thirds plus nine so nine is what 27 thirds so that leaves me with a negative eight thirds so negative eight thirds is the number there and then you do negative five thirds times one so that's going to be negative five thirds minus six that's negative five thirds minus what 18 thirds is minus 23 over 3.

01:38 Then let's look at, so then i would have what, 2 -1 -3 on the bottom? my next step would be to clear this number right here.

01:49 I could do that by saying what, negative 2 -thirds of row 1 and then add that to row 3 and place that answer back into row 3.

02:01 And so that would give me 371 -0.

02:09 Minus eight thirds, minus 23 thirds.

02:16 So negative two -thirds times three, and then add that to two, that gives me the zero i was looking for.

02:23 Negative two -thirds times seven.

02:26 So that's negative 14 -thirds minus, so negative two -thirds times seven.

02:35 Negative 14 -thirds minus eight -thirds is...

02:44 Oh, sorry.

02:51 Back up.

02:51 So negative two -thirds from seven, that's negative 14 -thirds plus 1.

02:55 So negative 14 -thirds plus 3 -thirds minus 11 -thirds is the number that goes here.

03:04 And then negative -thirds times 1, so that's negative 2 -thirds plus 3.

03:09 So that's negative 2 -thirds plus 9 -thirds, which is 7 -thirds...

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