Question

If all the diagonal entries of an $\mathit{n}\mathbf{\times }\mathit{n}$matrix $\mathit{A}$are odd integers and all the other entries are even integers, then$\mathit{A}$ must be an invertible matrix.

Short answer

Therefore, A is invertible. So, the given statement is true.

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

Using the definition of$detA$, the product in the addend corresponding to the diagonal pattern is odd, while all other addends are even.

Thus,$detA$is odd, so it must be different than 0.

Thus, $A$ is invertible. So, the given statement is true.