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Find (if it exists) a matrix $A$

of $R^{2X3}$,

with fist Row: $(0,6,6)$

so that zero is the unique solution of equation

$ A $ $ \begin{pmatrix} x \\ y \\ z\end{pmatrix}$ = $ \begin{pmatrix} 0 \\ 0 \end{pmatrix}. \\$

1 Answers 1

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This is impossible; such a matrix does not exist.

A 2 by 3 matrix has two rows and three columns, so its maximum rank is 2 and there is at least one free variable. That free variable, usually z, is replaced by a parameter that can take on any value in R.

Thus a unique solution cannot be found.

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    So, if we had a **3 by 2** matrix it would be the same, right? I mean if we know again the first line and we want a unique solution.... – 2017-02-05
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    A 3 by 2 matrix would be a whole different question. In that case you could only multiply vectors of length 2. A random 3 by 2 matrix equation is likely to be inconsistent if not homogeneous (right side zero); there is one more equation than variable, an excess of restrictions. The rank of a 3 x 2 matrix will be either 2 or 1 (or zero if trivial, all zeros). Work out a few examples for yourself and see how they fall out. – 2017-02-05
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    Well, the first row is given $(a,b)$ with $a,b\ne0$ so let $A=(\pmatrix{a=1&b=2\\c&d\\e&f})$ and, also, let $M=\pmatrix{1&2&0\\c&d&0\\e&f&0}$ $rankA=1 or 2$ and $rankM=1 or 2$ So matrix A does exist. For example.... $A=\pmatrix{1&2\\1&2\\1&1}$, right?? – 2017-02-06