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Chapter 6: Q 6.2-53E (page 292)

Question: Consider an invertible 2脳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

Expert verified

a) The given result is proved.

b) The A-1 is given by,

Step by step solution

01

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 鈥� m by n鈥� matrix, written 鈥� 尘脳苍鈥�

02

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.

03

To find  A-1

We have,

,with all integer entries.

If det A = 卤1, then

This again has all integer entries.

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Most popular questions from this chapter

a. Find a noninvertible 2脳2matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible 3脳3matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

5.[0234000412340034]

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

8. [0000210003010040010500016]

Find all 2脳2 matrices Asuch thatadj(A)=AT.

There exist real invertible 3脳3 matrices A and S such that .

See all solutions

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