Let A be a square matrix with integer entries and det A=1, and b a vector with integer entries. What can be said about the solution of Ax=b?

All its components are integers

At least one component is not an integer

It can be any rational number, with no further information

It does not exist, since det A must differ from 1

Exercise · ZW-CRAMERS-RULE-010 · ★★★ Advanced

Let \(A\) be a square matrix with integer entries and \(\det A=1\), and \(\mathbf{b}\) a vector with integer entries. What can be said about the solution of \(A\mathbf{x}=\mathbf{b}\)?

Choose one option.

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By Cramer's rule, \(x_i=\det A_i/1=\det A_i\), and the determinant of an integer matrix is an integer. Hence every \(x_i\) is an integer.

Cramer's rule gives the unique solution of a square system, whose coefficient matrix \(A\) has nonzero determinant, as a quotient of determinants.

\[x_i=\frac{\det A_i}{\det A}\]