Main content

### Course: Algebra (all content) > Unit 20

Lesson 11: Determinant of a 2x2 matrix# Determinant of a 2x2 matrix

Sal shows how to find the determinant of a 2x2 matrix. Created by Sal Khan.

## Want to join the conversation?

- What is the use of finding the determinant of a matrix?(160 votes)
- it helps in solving the system of linear equations(119 votes)

- what is exactly determinant of matrix? why does it exist and why do we calculate it like this?(76 votes)
- The determinant of a 2 by 2 matrix that is:

[a b]

[c d] is ad-cb . You can use determinants to find the area of a triangle whose vertices are points in a coordinate plane and you can use determinants to solve a system of linear equations. The method is called Cramer's Rule. In Cramer's rule you find the determinants of two matrices and divide them to find the x- and y-coordinate pair that solves the linear equation.(92 votes)

- This video seems to have a missing earlier step - the video talks about determinants without having previously introduced or defined them (in a previous (matrix) video?), and Sal appears to be continuing a previous (unseen?) lecture by starting "...as a hint if we take the determinant of a very similar 2x2 matrix..." So how does this link in with the previous section on geometric transformations? and where is the missing video(s)?(50 votes)
- Determinants originate as applications of vector geometry: the determinate of a 2x2 matrix is the area of a parallelogram with line one and line two being the vectors of its lower left hand sides. (Actually, the absolute value of the determinate is equal to the area.)

Extra points if you can figure out why. (hint: to rotate a vector (a,b) by 90 degrees, you turn it into (-b,a).

Honestly I don't know why Sal doesn't have any of this. It's just "here's some stuff! memorize it".(8 votes)

- What's the use in real life?(5 votes)
- You can use matrices to help you find all sorts of things! Like circulation, flux, gradient (slope), vectors, pretty much anything that has to do with physics, and when you get into calculus, matrices helps a TON with figuring out the change of variables in integrals... matrices are cool stuff!(11 votes)

- Is this video in the right place? Are there other videos intended to come before this one? The term "determinant" is supplied context free and devoid of any explanation of its origin, use, meaning or importance. Since I know that Sal provides context for all other concepts I can only conclude that there is another video that should be listed first. Where is it?(16 votes)
- can we find determinant of any non square matrix? if not why?(9 votes)
- No, only square matrices have determinants. One reason for this is that determinants are to do with the
**inverse**of a matrix (that is, the inverse of a matrix**A**is a matrix**B**such that**AB**=**BA**=**I**, for the appropriate identity matrix**I**). The equation**AB**=**BA**cannot exist for a non-square matrix because the rules of matrix multiplication would require the two B's to be of different sizes.(14 votes)

- Does this method apply to matrices of 3 by 3, 1 by 1, and so on?(4 votes)
- No. There are different methods for those. See:

https://www.khanacademy.org/math/algebra/algebra-matrices/inverting_matrices/v/finding-the-determinant-of-a-3x3-matrix-method-1(7 votes)

- What is the function of the determinant of a matrix? That is to say, what can I do with it?(2 votes)
- If the determinant is non-zero the matrix has a unique inversion, which means that if the matrix represents a system of linear equations, then the system also has a unique solution. So calculating the determinant can save you a lot of work trying to find a solution to a system of equations that has no solution.

For now, this is all you will be using the determinant for, BUT it has many more uses in more advanced math.(6 votes)

- So what exactly is a determinant?(3 votes)
- For two equations with two unknowns you put the constants into three matrices and when solved as determinants you can solve for X and Y rather easily. You can do the same by other means, so when I took algebra long ago, I think I shorted matrices. The same is true for equations of three or more unknowns.(1 vote)

- Are determinants only for 2x2 matrices?(2 votes)
- no. you can have determinants for 3x3, 4x4, 5x5 and so on(2 votes)

## Video transcript

As a hint, I'll take the determinant of a very similar two by two matrix. So we wanna figure out the determinant of this matrix and we can denote it by the determinant of the 2x2 matrix five, three,negative one and four and the definition of determinant of the 2x2 matrix like this, as we can first take this diagonal, five times four so write it down, just going to be equal to 5 * 4 minus of product of that diagonal ...so what is this give us? So, five times four is twenty and then three times negative one will be negative three this is negative three... but making negative the negative three will make the positive three so the determinant of this matrix is twenty three.