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Chapter 6: Q37E (page 290)

Consider n脳n a matrixA such that bothA andA-1 have integer entries. What are the possible values of detA?

Short Answer

Expert verified

Therefore, the possible values is 1or -1.

Step by step solution

01

Step by Step Solution: Step 1: Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

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

If there are mrows and ncolumns, the matrix is said to be a 鈥�m by n鈥� matrix, written 鈥�m脳n .鈥�

02

To find the possible values

If both Aand A-1have all integer entries, then both determinants are also integers.

We know that,

1=detIn1=detAA-11=detAdetA-1

which narrows detAdown to either -1or 1.

Therefore, the possible values is 1or -1.

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