Finding rank of augmented matrix

Question

Suppose $A$ is a $4 \times 4$ matrix with rank 3 and $I$ is the $4 \times 4$ identity matrix. How can I determine the rank of the augmented matrix

$$[A\,\, I]?$$

Could the rank be 4?

From $I$ it seems that we have four independent column vectors and since they form a basis the columns of $A$ are dependent on them. Does that argument seem right?

Answer 1

you can follow the normal approach of finding rank of a matrix by reducing it to either row reduceed echelon form or counting the nonzero rows in row reduceed form.I prefer the last.

As you give your matrix has rank 3 so it has 3 nonzero rows in row reduceed form and since you adding a identity matrix of order $4 ×4$ it is in the row reduceed form always.

Now when you come at the last row you have all element is zero of your original matrix and 1 in the identity matrix making the whole system having a nonzero fourth row . So your answer is correct and it is 4