Please prove from 1 to 4. Problem 4. (40 points) For each of the following four

Question

Please prove from 1 to 4.

Problem 4. (40 points) For each of the following four statements, determine whether the statement is TRUE or FALSE. If the statement is true, give a proof. If the statement is false, give a counterexample. You will receive NO credit if you do not give a proof or a counterexample. (1) A matrix M is invertible if and only if its transpose MT is invertible. (2) Let M be an n × n diagonal matrix. Then the dimension of the space of solutions of a homogeneous linear system of equations with coefficient matrix M is equal to the number of diagonal entries of MM that are equal to zero. (3) Let N be a k × n matrix of rank r. Let M be an m × k ma- trix. The matrix MN has rank r if and only if NullSpace(M)n ColumnSpace(N) = {0} sequence is called a short exact sequence of finite dimensional vector spaces if Vi are finite dimensional vector spaces and hi are homomorphisms such that hi : Vi ½ is one-to-one (injective), h2 : ½ VS is onto (surjective) and the kernel of h2 equals the image of h. For a short exact sequence of finite dimensional vector spaces, dim(V)-dim(½) + dim(V) = 0.

Explanation / Answer

Ans(1):
Let A is invertible matrix then there exists a matrix B such that AB=I, where I is the identity matrix.
take transpose of both sides
(AB)^T=I^T
B^T A^T =I^T {since (xy)^T=y^T x^T }
B^T A^T =I {since I^T=I }
then by uniqueness of inverse you have B^T is inverse of A^T.
Which proves that a matrix M is invertile if and only if it's transpose M^T is invertible.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.