Question

For some nonzero 2x2 matrix, with its entries in an integral domain, show that the matrix is a zero divisor IFF det=0. Show an example!

Answer #1

Show that there is no matrix with real entries A, such that A^2
= [ 0 1
0 0 ].
(its a 2x2 matrix)

Find an example of a nonzero, non-Invertible 2x2 matrix A and a
linearly independent set {V,W} of two, distinct
non-zero vectors in R2 such that
{AV,AW} are distinct, nonzero and
linearly dependent. verify the matrix A in non-invertible, verify
the set {V,W} is linearly independent and verify
the set {AV,AW} is linearly
dependent

Let R*= R\ {0} be the set of nonzero real
numbers. Let
G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
R*, b in R}
(a) Prove that G is a subgroup of GL(2,R)
(b) Prove that G is Abelian

Give an example of a nondiagonal 2x2 matrix that is
diagonalizable but not invertible. Show that these two facts are
the case for your example.

7) Let B be a matrix with a repeated zero eigenvalues. Then show
that B2 = 0 (the 2 × 2 zero matrix). Use this to show: if A has a
repeated eigenvalue λ0, then (A − λ0I) 2 = 0. (Hint: Use the fact
that Bv = 0 for some nonzero vector v)

please choose your favorite, unique 3x3 Matrix
A containing no more than two 0 entries and having a
nonzero determinant. I suggest choosing a matrix
with integer elements (e.g. not fractions or irrational numbers)
for computational reasons.
What is your matrix A? What is det (A)?
What is AT? What is det (AT)?
Calculate A AT. Show that A AT is
symmetrical.
Calculate AT A
Calculate the determinant of (A AT) and the
determinant of (AT A). Should the determinants...

Let B = [ aij ] 20×17 be a matrix with real entries. Let x be in
R 17 , c be in R 20, and 0 be the vector with all zero entries.
Show that each of the following statements implies the other.
(a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b)
If Bx = c has a solution for some vector c in R 20, then the
solution is unique.

Suppose A is a real 2x2 matrix with complex eigenvalues α ± i β
, β ≠ 0. It was shown in class that the corresponding eigenvectors
will be complex. Suppose that a + i b is an eigenvector for α + i β
, for some real vectors a , b . Show that a − i b is an eigenvector
corresponding to α − i β . Hint: properties of the complex
conjugate may be useful. Please show...

Show that A ∈ Mm,n(C) is the zero matrix if and only if all of
its singular values are zero.

Show that Thomae's function is Darboux/Riemann integrable and
its integral is equal to 0.

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