Question

If all the entries of an invertible matrixA are integers, then the entries of ${\mathit{A}}^{\mathbf{-}\mathbf{1}}$must be integers as well.

Short answer

The given statement is false.

Step-by-step solution

Matrix Definition

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

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 “$m\times n$ .”

To check whether the given condition is true or false

To find $A^{-1}$:

For example,

$A=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]$is an invertible matrix with all integer entries.

By using $A^{-1}$formula, however

$A^{-1}=\left[\begin{array}{cc}\frac{-1}{2}& \frac{3}{2}\\ \frac{1}{2}& \frac{-1}{2}\end{array}\right]$

Therefore, the A and$A^{-1}$ both are different.

So, the given statement is false.