Matrix Operations

[D. Wellington]

We've stepped into matrix operations this week and there are a few theorems that I don't quit understand. Now transposing a matrix is fairly straightforward and easy to comprehend, at least with small matrices. The part I'm having trouble with is the invertible matrix. I understand the key component that A^-1 is the inverse of A; however, I'm absolutely lost by the idea -- E_p...E₂, E₁A=I_n and the algebraic transformations of this equation. What is the purpose in finding the elementary matrix and what is a good way to visualize and understand this concept?

[ecoli]

We've stepped into matrix operations this week and there are a few theorems that I don't quit understand. Now transposing a matrix is fairly straightforward and easy to comprehend, at least with small matrices. The part I'm having trouble with is the invertible matrix. I understand the key component that A^-1 is the inverse of A; however, I'm absolutely lost by the idea -- E_p...E₂, E₁A=I_n and the algebraic transformations of this equation. What is the purpose in finding the elementary matrix and what is a good way to visualize and understand this concept?

 

 

You can think of matrix inversion as analogous to the inversion of any real number.

 

Just like: [math] 4 * 4^{-1} = 4 * \frac{1}{4} = 1 [/math]

 

For some matrices, A, there exists a so-called inverse such that:

 

[math] AA^{-1} = I [/math], I being the identity matrix - a square matrix with 1's along the diagonal and 0's everywhere else.

 

 

Elementary matrices allow you to perform operations on matrix A, for example, to solve for the inverse matrix.

[D. Wellington]

You can think of matrix inversion as analogous to the inversion of any real number.

 

Just like: [math] 4 * 4^{-1} = 4 * \frac{1}{4} = 1 [/math]

 

For some matrices, A, there exists a so-called inverse such that:

 

[math] AA^{-1} = I [/math], I being the identity matrix - a square matrix with 1's along the diagonal and 0's everywhere else.

 

 

Elementary matrices allow you to perform operations on matrix A, for example, to solve for the inverse matrix.

 

Thank you for your reply ecoli and I understand [math] A^-1 [/math]; however, I don't understand why it is necessary to do so --- in other words, what is the application of [math] A^-1 [/math]? Does the elementary matrix act as a "key" that unlocks a matrix, like something you might find in cryptography? It helps me to understand the topics in mathematics if I can understand where it might apply (real-world problem). For instance, does it allow us to construct a more detailed model of whatever is being tested?

[ecoli]

Thank you for your reply ecoli and I understand [math] A^-1 [/math]; however, I don't understand why it is necessary to do so --- in other words, what is the application of [math] A^-1 [/math]? Does the elementary matrix act as a "key" that unlocks a matrix, like something you might find in cryptography? It helps me to understand the topics in mathematics if I can understand where it might apply (real-world problem). For instance, does it allow us to construct a more detailed model of whatever is being tested?

 

That's a useful way to think of it. A practical application of matrix inverses are, because there's no definition of matrix division, multiplying by a matrix's inverse accomplishes the same thing. This is useful when you need to solve a system of linear equations.

 

Here's a problem that can be solved using inverse matrix multiplication:

 

A group is traveling by train and bus. Bus tickets cost $3 for children and $3.20 for adults. Train tickets cost $3.50 for children and $3.60 for adults. The group spent $118.40 on bus fare and $135.20 on train fare. How many adults and children were in the group?

 

 

 

You could imagine a really large and complex set of equations (which describes problems in engineering, physics, biology, computer graphics, etc) with a set variables you need to solve. Linear algebra provides a framework to do this by defining operations on matrices of numbers. One of those being matrix inversion, which is especially useful when a single solution exists to this set of equations (for example, only 1 combination of adults and children could have spent that combination on bus and train fare).