Recall that we have seen that an n times n matrix is invertible if and only if i
ID: 3036013 • Letter: R
Question
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.Explanation / Answer
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. AT is also invertible. The rows of A are columns of AT. Also, if A is invertible, then so is AT . 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.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.