Recall that we have seen that an n times n matrix is inverti

Recall that we have seen that an n times n matrix is invertible if and only if its RREF is the n times n identity matrix. Let A be an n times n matrix. For each of the following statements briefly(!) justify why they are true. A is invertible if and only if its columns are linearly independent. A is invertible if and only if its rows are linearly independent. A is invertible if and only if col(A) = R^n. A is invertible if and only if rank(A) = n. A is invertible if and only if nullity(A) = 0. A is invertible if and only if its columns form a basis of R^n. A is invertible if and only if row(A) = R^n.

Solution

A. If the columns are linearly independent, then det(A) 0 so that A is invertible. Also, if A is invertible, then det(A) 0 so that the columns are linearly independent.

B. A^T is also invertible. The rows of A are columns of A^T. Also, if A is invertible, then so is A^T . Then, by virtue of part A above, A is invertible if and only if the rows of A are linearly independent.

C. By virtue of part A above, A is invertible if and only if the columns of A are linearly independent i.e. if and only if the columns of A span Rⁿ.

D. By virtue of part B above, A is invertible if and only if the rows of A are linearly independent i.e. if and only if the rows of A span Rⁿ i.e. if and only if rank(A) = n.

E. As per the rank-nullity theorem, the rank +nullity of a matrix is equal to the number of columns of the matrix. By virtue of part D above, A is invertible if and only if rank(A) = n i.e. if and only if nullity(A) = 0.

F. By virtue of part A above, A is invertible if and only if the columns of A are linearly independent i.e. if and only if the the columns of A form a basis for Rⁿ.

G. By virtue of part B above, A is invertible if and only if the rows of A are linearly independent i.e. if and only if the the rows of A form a basis for Rⁿ i.e. row(A) = Rⁿ.