Matrices. Problem and answers.
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.
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.
