The augmented matrix of a linear system with three variables and four equations has rank 4. What can you say about the solutions of this system?

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Call the variables `x_1,x_2,x_3.` The augmented matrix is `4xx4.` It has rank 4, so by elementary row operations, it can be transformed into the `4xx4` identity matrix

`[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]].`

But the last row says that `0x_1+0x_2+0x_3=1,` which is impossible. Therefore there is no solution.

Another way to see this is to use the fact that the rank of a matrix is the size of the largest nonzero minor(s). Since this matrix is `4xx4` and has rank 4, this means that the determinant (which is the only `4xx4` minor) is nonzero. This in turn means that the columns are independent. In other words, the fourth column is not a linear combination of the first three. This means there is no solution.

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