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Thread: Why add or multiply matricies?

  1. #1
    Junior Member B9766's Avatar
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    Why add or multiply matricies?

    My current lesson is the second in the introduction to systems of linear equations and matrices. I understand how to use matrices to solve a system of linear equations and I understand how to add and multiply matrices.

    I don't understand why one would want to perform these functions. I tried graphing two systems of linear equations, identifying their solutions (intersection points). Then I added the associated matrices and graphed the resulting matrix. The solution of the result doesn't seem to have any meaningful relationship to the two original matrices nor their solutions.

    I understand the application of a single matrix to demonstrate Gaussian elimination as in the example:

    A football game has a total score of 39 points consisting of 3-point field goals, 6-point touchdowns and 1-point extra points. There were 11 plays in the game. There were the same number of touchdowns as field goals. How many of each type of score were there?

    But why then, would anyone multiply this system of linear equations by some other system? What could one hope to gain?

    I know this sounds esoteric but the next lesson is Inverses and Determinants of Matrices. I'd like to have an idea of the bigger picture before delving into the details.
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    Re: Why add or multiply matricies?

    In the example you give,
    [quote]A football game has a total score of 39 points consisting of 3-point field goals, 6-point touchdowns and 1-point extra points. There were 11 plays in the game. There were the same number of touchdowns as field goals. How many of each type of score were there?[/tex]

    Let x be the number of 3 point field goals, y be the number of 6 point touch downs, and z be the number of 1 point extra points. We reduce each sentence to an equation.

    "A football game has a total score of 39 points" 3x+ 6y+ z= 39.
    "There were 11 plays in the game" (I assume that should be "scoring plays) x+ y+ z= 11.
    "There were the same number of touchdowns as field goals" y= z which is the same as y- z= 0.

    We can write each equation as if it were a "dot product" of two vectors:
    3x+ 6y+ z= 39 is the same as (3, 6, 1).(x, y, z)= 39
    x+ y+ z= 11 is the same as (1, 1, 1).(x, y, z)= 11
    y- z= 0 is the same as (0, 1, -1).(x, y, z)= 0.

    But then that is the same as the matrix equation
    $\displaystyle \begin{bmatrix}3 & 6 & 1 \\ 1 & 1 & 1 \\ 0 & 1 & -1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}39 \\ 11 \\ 0 \end{bmatrix}$.

    That is, matrix multiplication is defined, with "dot product of each row with each column", to give each equation in the original set. And matrix addition is the same as adding coefficients of the same unknown to give the result of adding equations.
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    Junior Member B9766's Avatar
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    Re: Why add or multiply matricies?

    Thanks HallsofIvy. I solved it exactly the same way. I understand the mechanics of HOW to multiply matrices. My question was "WHY use matrix multiplication or addition at all?" In this example, Gaussian Elimination will yield the same results. What do we gain by using "dot product"?
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    Re: Why add or multiply matricies?

    I think what you have here is an understanding of the very basics of a field of mathematics without knowing where it's going to lead. There are more tools for you to learn in the manipulation of matrices and more applications of those tools. Some of these applications are difficult to achieve via other methods. For example, solving systems of differential equations such as \begin{align}\frac{\mathrm dx}{\mathrm dt} &= 3x + y + 2z \\ \frac{\mathrm dy}{\mathrm dt} &= 5x - 3y + z \\ \frac{\mathrm dz}{\mathrm dt} &= x + 5y - 4z \end{align} would be far from easy without the use of matrices.
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    Re: Why add or multiply matricies?

    Also you might like to google Leslie matrices, Markov chains, Leontief input-output models for some applications where matrices make life so much easier.

    You can't build a house until you learn how to use a hammer. Be patient, learn the basics well, and the time will come for some very interesting applications.
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    Re: Why add or multiply matricies?

    Quote Originally Posted by B9766 View Post
    Thanks HallsofIvy. I solved it exactly the same way. I understand the mechanics of HOW to multiply matrices. My question was "WHY use matrix multiplication or addition at all?" In this example, Gaussian Elimination will yield the same results. What do we gain by using "dot product"?
    Writing systems of equations allows us to treat systems of equations as single equations. The equation Ax= B, with A and B numbers is easy to solve: if A is not 0, so $\displaystyle A^{-1}$ exists, $\displaystyle x= A^{-1}B$. If A and B are matrices, A an invertible matrix, the solution is exactly the same: [tex]x= A^{-1}B[tex].
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