Why are linear equations usually written down as matrices?

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I've been taught that for any system of linear equations, it has a corresponding matrix.

Why do people sometimes use systems of linear equations to describe something and other times matrices?

Is it all just a way of writing things down faster or are there things you could do to matrices that you couldn't do to linear equations?
 

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  • #2
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I've been taught that for any system of linear equations, it has a corresponding matrix.

Why do people sometimes use systems of linear equations to describe something and other times matrices?

Is it all just a way of writing things down faster or are there things you could do to matrices that you couldn't do to linear equations?
Mostly matrices are a shorthand way of writing a system of linear equations, but there is one other advantage for certain systems : the ability to use a matrix inverse to solve the system.

For example, suppose we have this system:
2x + y = 5
x + 3y = 5

This system can be written in matrix form as:
##\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} 5 \\5 \end{bmatrix}##
Symbolically, the system is Ax = b, where A is the matrix of coefficients on the left, and b is the column vector whose entries are 5 and 5. (x is the column vector of variables x and y.)

Because I cooked this example up, I know that A has an inverse; namely ##A^{-1} = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}##
If I apply this inverse to both sides of Ax = b, I get ##A^{-1}Ax = A^{-1}b = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 5 \\5 \end{bmatrix}##
##= \begin{bmatrix} 2 \\1 \end{bmatrix}##

From this I see that x = 2 and y = 1. You can check that this is a solution by substituting these values in the system of equations.
 
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  • #3
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Mostly matrices are a shorthand way of writing a system of linear equations, but there is one other advantage for certain systems : the ability to use a matrix inverse to solve the system.

For example, suppose we have this system:
2x + y = 5
x + 3y = 5

This system can be written in matrix form as:
##\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} 5 \\5 \end{bmatrix}##
Symbolically, the system is Ax = b, where A is the matrix of coefficients on the left, and b is the column vector whose entries are 5 and 5. (x is the column vector of variables x and y.)

Because I cooked this example up, I know that A has an inverse; namely ##A^{-1} = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}##
If I apply this inverse to both sides of Ax = b, I get ##A^{-1}Ax = A^{-1}b = \frac 1 5 \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 5 \\5 \end{bmatrix}##
##= \begin{bmatrix} 2 \\1 \end{bmatrix}##

From this I see that x = 2 and y = 1. You can check that this is a solution by substituting these values in the system of equations.
I see, thank you very much!
 
  • #4
HallsofIvy
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Essentially, matrices allow you to write any system of linear equations as the single equation "Ax= b", the simplest form.
 
  • #5
FactChecker
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The shorthand notation provided by the matrix is very beneficial. Keeping track of the variables that the matrix operates on often clutters up the calculations. If you compose a sequence of linear operations ( E = A * B * C * D ), you can do the matrix manipulations easily. If you try to name and keep track of all the intermediate values, it is just an unnecessary mess. ( x2 = Dx1; x3 = Cx2; x4 = Bx3; x5 = Ax4; so x5 = E x1 )
 

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