welcome. i found this course being technical to many students in universities and colleges.
this summary is very vital for revision purposes as it will enable one to gather the information and content eastly.
it involves simple steps with detailed information. feel free to interrogate where you ...
Definition
Let AA be an n×nn×n matrix.
1. AA is nonsingular if the only solution to Ax=0Ax=0 is the
zero solution x=0x=0. Ie the determinant is equal to zero.
Summary
Let AA be an n×nn×n matrix.
1. If AA is nonsingular, then ATAT is nonsingular.
2. AA is nonsingular if and only if the column vectors of AA are
linearly independent.
3. Ax=bAx=b has a unique solution for
every n×1n×1 column vector bb if and only if AA is
nonsingular.
=solution
Problems
1. Determine whether the following matrices are nonsingular or not.
(a) A=⎡⎣⎢12101012−1⎤⎦⎥A=[10121210−1].
(b) B=⎡⎣⎢214101214⎤⎦⎥B=[212101414].
2. Consider the matrix M=[13412]M=[14312].
(a) Show that MM is singular.
(b) Find a non-zero vector vv such that Mv=0Mv=0, where 00 is
the 22-dimensional zero vector.
3. Let AA be the following 3×33×3 matrix.
A=⎡⎣⎢101111−12a⎤⎦⎥.A=[11−101211a].
Determine the values of aa so that the matrix AA is nonsingular.
, 4. Determine the values of a real number aa such that the
matrix A=⎡⎣⎢3200318aa0a+1⎤⎦⎥A=[30a230018aa+1] is
nonsingular.
5. (a) Suppose that a 3×33×3 system of linear equations is
inconsistent. Is the coefficient matrix of the system nonsingular?
(b) Suppose that a 3×33×3 homogeneous system of linear
equations has a solution x1=0,x2=−3,x3=5x1=0,x2=−3,x3=5. Is
the coefficient matrix of the system nonsingular?
(c) Let AA be a 4×44×4 matrix and
let v=⎡⎣⎢⎢⎢1234⎤⎦⎥⎥⎥ and w=⎡⎣⎢⎢⎢4321⎤⎦⎥⎥⎥v=[1234] and w=[
4321]. Suppose that we have Av=AwAv=Aw. Is the
matrix AA nonsingular?
6. Let AA be a 3×33×3 singular matrix. Then show that there exists
a nonzero 3×33×3 matrix BB such that
AB=O,AB=O,
where OO is the 3×33×3 zero matrix.
7. Let AA be an n×nn×n singular matrix. Then prove that there exists a
nonzero n×nn×n matrix BB such that AB=OAB=O, where OO is
the n×nn×n zero matrix.
8. Let v1v1 and v2v2 be 22-dimensional vectors and let AA be
a 2×22×2 matrix.
(a) Show that if v1,v2v1,v2 are linearly dependent vectors, then the
vectors Av1,Av2Av1,Av2 are also linearly dependent.
(b) If v1,v2v1,v2 are linearly independent vectors, can we conclude
that the vectors Av1,Av2Av1,Av2 are also linearly independent?
(c) If v1,v2v1,v2 are linearly independent vectors and AA is
nonsingular, then show that the vectors Av1,Av2Av1,Av2 are also
linearly independent.
9. Let AA be an n×nn×n matrix. Suppose that the sum of elements in
each row of AA is zero. Then prove that the matrix AA is singular.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller cherrygracious. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $11.09. You're not tied to anything after your purchase.
