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Precalculus
Course: Precalculus > Unit 7
Lesson 7: Matrices as transformations of the plane- Matrices as transformations of the plane
- Working with matrices as transformations of the plane
- Intro to determinant notation and computation
- Interpreting determinants in terms of area
- Finding area of figure after transformation using determinant
- Understand matrices as transformations of the plane
- Proof: Matrix determinant gives area of image of unit square under mapping
- Matrices as transformations
- Matrix from visual representation of transformation
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Intro to determinant notation and computation
Matrix determinants are easy to define and hard to understand. So let's start with defining them and introducing related notation. In other videos we will learn what they mean and how to use them. Created by Sal Khan.
Want to join the conversation?
Can you only find matrix determinants in 2x2 matrices?
If not, then what method would you use to find the matrix determinant in matrices with different dimensions?(3 votes)- It is actually possible to find determinants in other matrices (not 2 by 2). For example, if you wanted to find the determinant of a 1 by 1 matrix, the answer would just be that same number. However, if you were to find a matrix in the form of 2 by 3, 3 by 5, etc..., it would be impossible to find the determinant. As long as long as you are looking at a matrix in the form of x by x, where both values of x are equivalent, it is possible to find the determinant. However, from matrix to matrix in that form, finding the determinant varies.(3 votes)
But what is the determinant?(2 votes)- A determinant is essentially the scalar quantity which is found in a matrix obtaining the form of x by x, where both values of x are the same (square matrix).(3 votes)
How do you calculate the determinant of a matrix bigger than 2x2?(1 vote)
To compute the determinant of a matrix larger than 2x2, you can use a process called expansion by minors or cofactor expansion. Here are the general steps:
Choose any row or column of the matrix.
For each element in that row or column, compute its corresponding minor, which is the determinant of the submatrix obtained by deleting the row and column that the element is in.
For each element in that row or column, multiply it by its corresponding minor and its sign, which is positive or negative depending on its position in the matrix. Specifically, the sign of an element in row i and column j is (-1)^(i+j).
Sum up all the products obtained in step 3 to get the determinant of the original matrix.
This process may look daunting for larger matrices, but it can be simplified by choosing a row or column that has many zeros or that has a repeated pattern. Additionally, there are some properties of determinants, such as linearity and multiplicativity, that can make the computation easier in some cases.(2 votes)
Could anyone tell me why we calculate determinants diagonally? Is there a reason, I didn't see Sal talks about it?(1 vote)
Check out The Essence of Linear Algebra by 3Blue1Brown, the best series out there that will explain any doubt on the topic(2 votes)
Video transcript
- [Instructor] In this
video, we're gonna talk about something called
determinants of matrices. So I'll start just
telling you the notation and how do you compute it. And then we'll think about
ways that you can interpret it. So let's give ourselves
a two by two matrix here. So, and actually, I'll
give it in general terms. So let's say that this
top-left term here is A, and then this one here is B the top-right. The bottom-left is C, and then let's call this bottom-right D. And I'm gonna do that
in a different color. So this is D right over here. the determinant of this matrix. And actually, let me
just call this matrix. Let's say that this is matrix A. So there's a bunch of ways
to call the determinant, or have the notation for the determinant. We could write it like this. We could have these little, it looks like absolute value signs, but it really means determinant when you apply it to a matrix. So the determinant of matrix A. You can write it that way. You could write it this way,
the determinant of matrix A. You could write it that way. Or you could write it this way, where you put these lines that look like big absolute value
signs instead of the brackets when you describe the numbers. So you could also write it this way. And I haven't explained
what determinant is or even how to compute it yet. I'm just talking about the notation of how you even talk about
the determinant of a matrix. So you can also write it this way, just rewrite the whole matrix with those vertical bars next to it. This is defined as, and we'll see how it's
useful in the future, the top-left time's the bottom-right. So A times D minus the top-right times the bottom-left. BC. So another way to think about it, it is just these two, the
product of these two minus, so that's those two right over there, minus the product of
these two right over here. So let's just first, before
we start to interpret this, get a little practice, just
computing a determinant. So let me give you a matrix. So let's say I have the matrix one, negative two, three, and five. Pause this video and
see if you can compute the determinant of this matrix. Let's call this matrix B. I want you to figure out
the determinant of matrix B. What is this going to be equal to? All right, now let's do this together. So you're going to have the
product of these two numbers. So we have one times five minus the product of these two numbers, which is three times negative two. And that of course, is going to be equal to one times five is five, three times negative two is negative six. But we're subtracting a negative six. Five minus negative six is the
same thing as five plus six which is going to be equal to 11. Now that we know how to
compute a determinant, in a future video, I will give you an
interesting interpretation of the determinant.