Question

Show that a square matrix P over the integers has an inverse with integer entries if and only if P is unimodular, that is, the determinant of P is ±1.

Answer #1

Show that if a square matrix K over Zp ( p prime) is
involutory ( or self-inverse), then det K=+-1
(An nxn matrix K is called involutory if K is invertible and
K-1 = K)
from Applied algebra
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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 A be a square matrix with an inverse A-1.
Show that if Ab = 0 then b must be the zero vector.

A stochastic matrix is a square matrix A with entries 0≤a_ij≤1
such that the sum of each column of A is 1. Prove that if A is
stochastic, then A^k is stochastic for every positive integer
k.

A) Find the inverse of the following square matrix.
I 5 0 I
I 0 10 I
(b) Find the inverse of the following square matrix.
I 4 9 I
I 2 5 I
c) Find the determinant of the following square matrix.
I 5 0 0 I
I 0 10 0 I
I 0 0 4 I
(d) Is the square matrix in (c) invertible? Why or why not?

(a) Find the inverse of the following square
matrix.
I 5 0 I
I 0 10 I
(b) Find the inverse of the following square
matrix.
I 4 9 I
I 2 5 I
(c) Find the determinant of the following square
matrix.
I 5 0 0 I
I 0 10 0 I
I 0 0 4 I
(d) Is the square matrix in (c) invertible? Why or why
not?

Let A be a square matrix, A != I, and suppose there exists a
positive integer m such that Am = I. Calculate det(I + A
+ A2+ ··· + Am-1).

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

Argue that the only way for a square matrix Ain reduced echelon
form Arr to have a non-zero determinant is if Arr=I, the identity
matrix.

(1) A square matrix with entries aj,k , j, k = 1,
..., n, is called diagonal if aj,k = 0 whenever j is not equal to
k. Show that the product of two diagonal n × n-matrices is again
diagonal.

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