nn SolutionWe know that If A is an n n matrix such that A2

Solution

We know that If A is an n × n matrix such that A² = A and rank(A) = n then det(A) 0 ( as the rank can also be defined as the size of the largest non-zero determinant of a matrix). Hence A is invertible. Multiplying the given relation with A^-1 on the left, we get A = I_n.

In view of the above, if A² = A and A I_n, then Rank (A) = R(A) cannot be n, i.e. R(A) < n. Since the column rank and the row rank of a matrix are equal, hence the RREF of A will contain at least 1 zero column. This means that the columns of A cannot be linearly independent if A² = A and A I_n.