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

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A 3x3 matrix of linear forms cannot always be expressed as the sum of at most three rank one matrices of linear forms, a limitation not present when the entries are scalars. This discrepancy arises from the differences in calculating 2x2 minors for linear forms compared to scalars. For instance, the matrix with entries s, t, and zeros cannot be decomposed into three rank one matrices of linear forms, highlighting the complexity introduced by the linear nature of the entries.

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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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