Question

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).

Short answer

The command that will create a\(6 \times 4\)matrix with random entries is\( > > rand\left( {6,4} \right)\). The range of the entries is between 0 and 1.

The command \( > > round\left( {19*\left( {rand\left( {3,3} \right) - .5} \right)} \right)\) creates a \(3 \times 3\) matrix with random integer entries between \( - 9\) and 9.

Step-by-step solution

Write the MATLAB commands

To create a matrix of the order\(m \times n\)with random entries, use the command\( > > rand\left( {m,n} \right)\), and distribute the entries between 0 and 1.

To create a matrix of the order\(m \times n\)with integer entries between\( - a < b\left( {x - c} \right) < a\), use the command\( > > round\left( {b*\left( {rand\left( {m,n} \right) - c} \right)} \right)\)and distribute the integer entries between\( - a\)and a.

Create a \(6 \times 4\) matrix with random entries

To create a matrix of the order\(6 \times 4\)with random entries, use the command in the MATLAB, as shown below:

\( > > rand\left( {6,4} \right)\)

The obtained random\(6 \times 4\)matrix is shown below:

\(\left( {\begin{aligned}{*{20}{c}}{0.78172}&{0.57396}&{0.54561}&{0.08579}\\{0.45554}&{0.76801}&{0.56704}&{0.91465}\\{0.44674}&{0.39120}&{0.15160}&{0.02506}\\{0.79187}&{0.83126}&{0.97255}&{0.05501}\\{0.47461}&{0.14454}&{0.44708}&{0.76701}\\{0.86106}&{0.45261}&{0.50968}&{0.08317}\end{aligned}} \right)\)

The entries of the matrix are in the interval\(\left( {0,1} \right)\).

Thus, the range of the entries lies between 0 and 1.

Create a \(3 \times 3\) matrix with random integers

To create a matrix of the order\(3 \times 3\)with random integers between\( - 9\)and 9, assume x as a random number represented as\(0 < x < 1\).

Then,\( - 9.5 < 19\left( {x - .5} \right) < 9.5\).

Use the command in the MATLAB as shown below:

\( > > round\left( {19*\left( {rand\left( {3,3} \right) - .5} \right)} \right)\)

Thus, the command is \( > > round\left( {19*\left( {rand\left( {3,3} \right) - .5} \right)} \right)\).