Say the system is

`a_{11}x+a_{12}y+a_{13}z=b_1`

`a_{21}x+a_{22}y+a_{23}z=b_2`

`a_{31}x+a_{32}y+a_{33}z=b_3`

`a_{41}x+a_{42}y+a_{43}z=b_4,`

so the augmented matrix is

`[[a_{11},a_{12},a_{13},b_1],[a_{21},a_{22},a_{23},b_2],[a_{31},a_{32},a_{33},b_3],[a_{41},a_{42},a_{43},b_4]].`

For this to have rank 4, the columns must be independent, so no linear combination of the first three columns can result in the fourth column. In other words, there is no solution to the system.

...

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Say the system is

`a_{11}x+a_{12}y+a_{13}z=b_1`

`a_{21}x+a_{22}y+a_{23}z=b_2`

`a_{31}x+a_{32}y+a_{33}z=b_3`

`a_{41}x+a_{42}y+a_{43}z=b_4,`

so the augmented matrix is

`[[a_{11},a_{12},a_{13},b_1],[a_{21},a_{22},a_{23},b_2],[a_{31},a_{32},a_{33},b_3],[a_{41},a_{42},a_{43},b_4]].`

For this to have rank 4, the columns must be independent, so no linear combination of the first three columns can result in the fourth column. In other words, there is no solution to the system.

This is a special case of the theorem described in the link, which says that a system of equations has a solution if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. Here, the largest rank the coefficient matrix can have is 3 (since there are 3 columns), so there can be no solution.

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