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Mathematicians in many disciplines like to construct rectangular arrays
of formulas that have been arranged in rows and columns; such an array
is called a {\it matrix}. Plain \TeX\ provides control sequences that
make most common types of matrices easy to print.
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You can print simple matrices to indicate their product:
$$\left(\matrix{a&b&c\cr
                  d&e&f\cr}\right) \left(\matrix{u&x\cr
                                                 v&y\cr
                                                 w&z\cr}\right).$$
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More complex matrices are just as easy to print:
$$A=\left(\matrix{x-\lambda&1&0\cr
                   0&x-\lambda&1\cr
                   0&0&x-\lambda\cr}\right).$$
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Even genertic patterned matrices that use ellipses to indicate rows or columns
are simple to set up and print:
$$A=\pmatrix{a_{11}&a_{12}&\ldots&a_{1n}\cr
             a_{21}&a_{22}&\ldots&a_{2n}\cr
             \dots&\vdots&\ddots&\vdots\cr
             a_{m1}&a_{m2}&\ldots&a_{mn}\cr}.$$
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Determinants are constructed in much the same way with the same ease:
$$\det\left|\,\matrix{
               c_0&c_1\hfill&c_2\hfill&\ldots&c_n\hfill\cr
               c_1&c_2\hfill&c_3\hfill&\ldots&c_{n+1}\hfill\cr
               c_2&c_3\hfill&c_4\hfill&\ldots&c_{n+2}\hfill\cr
               \,\vdots\hfill&\,\vdots\hfill&\,\vdots\hfill&&\,\vdots\hfill\cr
               c_n&c_{n+1}\hfill&c_{n+2}\hfill&\ldots&c_{2n}\hfill\cr}\right|>
0.$$
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