Question

Consider an nxn matrix A with integer entries such that A=1. Are the entries of${\mathbf{A}}^{\mathbf{-}\mathbf{1}}$ necessarily integers? Explain.

Short answer

Yes, the entries of${\mathrm{A}}^{-1}$is the necessarily integers.

Step-by-step solution

Matrix Definition. 

Matrix is a setof 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 m rows and n columns, the matrix is said to be an “m by n” matrix, written “mxn”

Step 2: Are the entries of  necessarily integers.

Since A has all integer entries, thenmust also have all integer entries.

If det A = 1, then

$A^{-1}=\frac{1}{detA}adjA$also has all integer entries.