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## Precalculus

### Unit 7: Lesson 14

Finding inverses of 2x2 matrices

# Finding inverses of 2x2 matrices

Sal gives an example of how to find the inverse of a given 2x2 matrix. Created by Sal Khan.

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• What if the determinant is 0? Is the answer undefined? •   When the determinant for a square matrix is equal to zero, the inverse for that matrix does not exist.
So yes, there is no inverse if the determinant is 0.
• Shouldn't A and A' be separeted from each other when writing the equations? Cause A couldn't be equal to its inverse...? •   You are right. After he wrote A=, he should have started a new line and wrote A'=....and then continued with everything he does. However, it's clear what Sal is doing if you listen to what he is saying, but it would get confusing if you were just looking at the equations.
Applied mathematicians (physicists, computer scientists, engineers etc) are often sloppy with mathematical notation. The emphasis is often on getting the result rather than on the rigorous proof of each step. If you search elsewhere online, you will find that there is a lot of criticism for Khan Academy for its inconsistant use of proper mathematical notation. (But imperfect as it may be, it still is making math accessible to lots of people)
• • What Sal introduced here in this video, is a method that was 'woven' specially for finding inverse of a 2x2 matrix but it comes from a more general formula for determining inverse of any nxn matrix A which is:
where adj(A) - adjugate of A - is just the transpose of cofactor matrix Cᵀ.
Cofactor matrix C of matrix A is also nxn matrix whose each entry (Cᵢ,ⱼ for example) is the determinant of the submatrix formed by deleting the i-th row and j-th column from our original matrix A multiplied by (-1)^(i+j).

Saying all of that, let's try it on 3x3 matrix. Suppose we have a matrix B
⌈a b c⌉
| d e f |
⌊g h i ⌋

the first thing is to find determinant of B:
det(B) = a(ei - fh) - b(di - fg) + c(dh - eg)

second thing is to find the cofactor matrix C of B (which takes a lot of effort):
⌈C₁,₁ = (-1)² * (ei - fh) C₁,₂ = (-1)³ * (di - fg) C₁,₃ = (-1)⁴ * (dh - eg) ⌉
|C₂,₁ = (-1)³ * (bi - ch) C₂,₂ = (-1)⁴ * (ai - cg) C₂,₃ = (-1)⁵ * (ah - bg)|
⌊C₃,₁ = (-1)⁴ * (bf - ec) C₃,₂ = (-1)⁵ * (af - cd) C₃,₃ = (-1)⁶ * (ea - bd)⌋

finally, we transpose our last matrix C to get the adjugate of B:
⌈(ei - fh) (-bi + ch) (bf - ec) ⌉
|(-di + fg) (ai - cg) (-af + cd)|
⌊(dh - eg) (-ah + bg) (ea - bd)⌋

so B⁻¹ = 1/det(B) * adj(B)
usually, you let the computer calculate the inverses for you.
• • You can add, subtract, and multiply matrices. Division, however, is not defined. Thus to undo matrix multiplication, you need to multiply by the inverse matrix. It is thus a pretty fundamental operation.

One early application for inverse matrices is to solve systems of linear equations. You can express the system as a matrix equation AX=B, then solve it by multiplying by the inverse of the coefficient matrix to get X = A^(-1)*B
• • Do you like Pixar films?
Matrix math is used intensively in making 2D and 3D graphics - they have developed special chips just to do this type of math as fast as possible. Back when the original Toy Story came out, it took 24 hours just to do the math (using matrices) to render one frame of animation.
Matrices are also a great way to solve simultaneous systems of linear equations. These are used in as many different fields as you can think of, from medicine and all its applications, finances, forecasting, trend analysis, system optimization and on and on.
Since there is no direct way to do matrix division, we take the inverse of a matrix and then multiply the matrix we want to divide by the inverse, just like 3÷2 = 3×½.
• • Is there a video on Khan Academy where Sal proves the inverse matrix equation? If there is, I'd like to know.  •  