Question

Question: Consider an invertible 2×2 matrix Awith integer entries.
a. Show that if the entries A^-1 of are integers, then det A = 1 ordetA= -1.
b. Show the converse: If detA = 1 ordetA =-1, then the entries of are integers.

Short answer

a) The given result is proved.

b) The A^-1 is given by,

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged inrowsandcolumnsso as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a “ m by n” matrix, written “ m×n”

To show  det  A =  ± 1

If all entries of both A and a^-1 are integers, then both determinants are integers, too. We have.

Then, detA^-1 can only be -1 or 1.

To find  A-1

We have,

,with all integer entries.

If det A = ±1, then

This again has all integer entries.