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

Elementary Row Operations

Elementary row operations include swapping rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another row. These operations are used to simplify matrices, such as converting them to upper triangular form, and each type of operation has a specific effect on the determinant. For example, swapping rows changes the sign of the determinant, while multiplying a row by a constant multiplies the determinant by that constant.

Upper Triangular Matrix

An upper triangular matrix is a matrix in which all the entries below the main diagonal are zero. One of the key properties of an upper triangular matrix is that its determinant is simply the product of its diagonal entries. This property is especially useful in simplifying the computation of determinants after transforming a matrix into an upper triangular form using row operations.

Determinant

The determinant is a scalar value that is computed from a square matrix and encodes important properties of the matrix such as invertibility. It can be used to solve systems of linear equations, find the inverse of a matrix, and understand the volume scaling factor associated with the linear transformation represented by the matrix.

Transcript

00:02 Section 3 .2, problem 10, we're dealing with properties of determinants.

00:06 Here we're presented with a 4 by 4 matrix.

00:09 We're asked to use elementary row operations to get into upper triangular form.

00:14 So that means all zeros below the diagonal.

00:16 So the determinant would be the product of all the numbers that are on the diagonal.

00:21 So let's just methodically start working through this.

00:24 I just go column by column.

00:26 So the first thing i want to do is let's clear this number you see right here.

00:32 So in order to do that, if i took negative three halves times row 1 plus row 2 and replace that into row 2, that would clear the number you see there.

00:46 So that would give me 2, 1, 3, 5.

00:53 And then negative 3 halves times 2 is negative 3.

00:57 Add to 3, you get 0.

00:58 0.

01:00 Negative 3 halves plus 1 is negative 3 halves add to 0.

01:04 You get negative 3 halves.

01:09 Negative 3 halves and 3 is negative 9 halves.

01:11 Negative 9 halves plus 1, that's negative 9 halves plus 2 halves is negative 7 halves.

01:20 Negative 3 halves times 5 is negative 15 halves.

01:23 So negative 15 halves plus 2, negative 15 halves plus 2, negative 15 halves plus 4 halves, is negative 11 halves.

01:30 And then the last three, last two rows remain unchanged.

01:41 Next thing i want to target is the item you see there.