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Let \m and \n be two positive integers. A \m&times'\n <em>matrix</em>
is a table of \m\n elements arrayed in \m lignes and \n colonnes. 
The matrix is <em>real</em> if the elements are real numbers, or
<em>complex</em> if they are complex numbers, etc.

For example, \([1,pi;e,-3]) is a real 2&times;2 matrix, and
\([1,i,3;4,5,pi-i;-i,-1,-1]) is a complex 3&times;3 matrix.

One may write a general \m&times;\n matrix by
 <div class="wimscenter">
\embed{matmn}{A&nbsp;=&nbsp;}
</div>
or abridged to (\a<sub>\i\j</sub>). The \a<sub>\i\j</sub>
are called <em>coefficients</em> of A, and \m&times;\n is its <em>dimension</em>.

Two matrices A and B are equal only if they are of the same dimension,
and have the same coefficient at every corresponding position.

\fold{matdefs}

To calculate a matrix, here is an online tool.
\calcform{alg/lin/matrix}

The \link{matdefs,trace,trace} of matrices satisfies the formula
 <div class="wimscenter">
   \(trace(AB)=trace(BA))
</div>
when \(AB) and \(BA) both make sense.

\fold{Preuve_trace_comm}

some exercises on matrices:
\exercise{cmd=new&module=U1/algebra/oefmatrix.en}&nbsp;
\exercise{cmd=new&session=robot&module=U1/algebra/dialmatrix.en}