Question

a. Find a noninvertible $\mathbf{2}\mathbf{\times }\mathbf{2}$matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible $\mathbf{3}\mathbf{\times }\mathbf{3}$matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

Short answer

a. No such matrix exists.

b.$\left[\begin{array}{ccc}7& 2& 11\\ 17& 3& 23\\ 19& 5& 19\end{array}\right]$

Step-by-step solution

Matrix Definition

Matrix is a set 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 “$m\times n$.”

(a) To find the noninvertible 2×2 matrix

Recall that for the general $2\times 2$matrix's determinant is calculated with the formula:

$detA=\mathrm{de}t\left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=ad-bc$

Since a,b,c and d are distinct prime numbers, so$\mathrm{detA}\ne 0$

Therefore, such matrix exists.

(b) To find the noninvertible 3×3 matrix

A $3\times 3$matrix is not invertible if one of it's columns is an linear combination of another two. We can arrange the two first columns to be linearly independent and we can attempt to find a linear combination of the two that will yield the third column. The only condition is that the new column is full of prime numbers. This can be arranged in numerous ways, one example is given below:

$\left[\begin{array}{ccc}7& 2& 11\\ 17& 3& 23\\ 19& 5& 19\end{array}\right]$

In the example given, the third column is equal to the sum of the second column multiplied by two and the first column.