Can a matrix of linear forms always be written as the sum of rank one matrices?

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Why is it a (for example) 3x3 matrix of linear forms cannot necessarily be written as the sum of at most 3 rank one matrices of linear forms but the statement is true if "linear forms" is replaced with scalars? Does it have something to do with the 2x2 minors being calculated differently when the entries are linear forms versus scalars? For example:

s t 0
0 s t
0 0 s

cannot be written as the sum of 3 rank one matrices of linear forms.
 
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