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If a matrix can represent a system of equations, what is the meaning of the square of that matrix? It represents another system? What is the relation with the final system and the first?

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A matrix can represent things other than a system of equations. –  Qiaochu Yuan Jul 17 '12 at 20:16
    
Note that a matrix represents about a class of linear equations, not just one set of linear equations. A class of problems of the form $Ax = b$ where $b$ can e any vector. Also note that the square of $A$ only makes sense for that class of problems if the set of equations and unknowns represented by $Ax=b$ are equal. –  Thomas Andrews Jul 17 '12 at 20:32
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I'm not sure if this clarifies or obscures.

Write your set of equations as:

$$Ax = b$$

Where $b$ is a vector and $A$ is an $n\times n$ matrix.

Then solving

$$A^2 x = b$$

Is the same as solving:

$$A x = y$$ $$A y = b$$

Therefore, if we think of the $y$ as another $n$ unknowns, we can write the $2n\times 2n$ matrix for this set of $2n$ equations in $2n$ unknowns as:

$$\left(\begin{array}{rrr} A & -I \\ 0 & A \\ \end{array}\right)\left(\begin{array}{rrr} x\\ y \end{array}\right) = \left(\begin{array}{rrr} 0\\ b \end{array}\right)$$

That this "reduces" to solving an equation of the form $A^2 x = b$ is the property of $A^2$.

We can actually do much the same thing for any product of two matrices.

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This is the answer I was looking for. Thanks! –  dot dot Jul 17 '12 at 21:24
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When we think of a matrix as a linear transformation, the square of a matrix $A$ is the transformation defined by applying $A$ twice. Keep in mind that $A^2$ is only defined when $A$ is square (not to be confused with squaring). This is to say that we can only apply a transformation again if we are mapping from a space into itself.

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Even in the case of a square matrix $A$ (so that $A^2$ makes sense) you cannot expect to understand $A^2$ in terms of the interpretation of the matrix as a system of equations $A\cdot x=0$ (note that it cannot represent a system $A\cdot x=b$ because such $b$ is not contained in the matrix). That interpretation views each row of $A$ as an equation. But in a system of equations the order of the equations does not affect the meaning of the system, so this meaning is unchanged under permutation of the rows. However such permutation can change the value of $A^2$ in a dramatic way, so $A^2$ cannot be understood in terms of the system $A\cdot x=0$ alone.

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