Find the value of | 30 11 20 38 | 6 3 0 9 | 11 -2

step 1 Hi guys, let's solve problem 14. Here is the determinant. Here we can see that this is a triangu

Find the value of | 30 11 20 38

Key Concepts

Determinant

The determinant is a scalar value derived from a square matrix that encapsulates important properties such as whether the matrix is invertible and how it scales area or volume when viewed as a linear transformation. It is used in various branches of mathematics, particularly in linear algebra, to study systems of linear equations, eigenvalues, and transformations.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method to compute the determinant of a matrix by breaking it down into determinants of smaller submatrices (minors). Each element of a row or column is multiplied by its corresponding cofactor, and the sum of these products gives the overall determinant. This concept is fundamental for understanding the recursive nature of determinant calculation.

Row Reduction

Row reduction is a technique used to simplify the calculation of determinants by applying elementary row operations to transform a matrix into a form that is easier to evaluate, such as an upper triangular matrix. In this form, the determinant is simply the product of the diagonal elements. Understanding the effects of row operations on the determinant, including any necessary sign changes or scalar factors, is crucial in this approach.

Transcript

00:01 Hi guys, let's solve problem 14.

00:04 Here is the determinant.

00:07 Here we can see that this is a triangular determinant as here is a triangle and after that here all the elements are zero.

00:21 So in this case we can evaluate this in two ways.

00:27 So here we can take the first column as it contains.

00:34 Highest number of 0 so it will be easier to calculate so let's take negative 2 negative 2 and after canceling the row and column we get this 3 by 3 matrix and rest of the items are 0 so we can write it like this but we don't need to calculate this because whatever we multiply with 0 the result will be 0 so we can cancel out these three parts.

01:04 Now, let's calculate the first part.

01:08 So here, here is a 3 by 3 matrix.

01:12 Let's calculate it.

01:14 So, the first row of this 3 by 3 matrix is here.

01:19 So the first number here is 3.

01:21 So we will write 3.

01:23 And after canceling the row and column of 3, we get this 2 by 2 matrix.

01:28 Now we will write the second number which is 0 we are considering the first column sorry we are considering the first column as it has highest number of 0 so the calculation will be smaller so here the second number is 0 so negative 0 and then after canceling the row and column we will get this 2 by 2 matrix in the same way we will take the third number 0 and cancel out the row and column to get this 2 by 2 metrics.

02:04 But here same again, we don't have to calculate these parts as it contains 0.

02:09 So whatever we multiply with 0, the result will be 0...

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