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Q58 E

Expert-verifiedFound in: Page 277

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**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.**

a. No such matrix exists.

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

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.

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.

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