Singular Matrices The following code generates 100 2 2 matr

Singular Matrices

The following code generates 100 2 × 2 matrices with integer coefficients in the ranges k = 1, 2, … , 20. For each fixed k, it finds the percent that are singular.

percent = zeros(1,20)\'; for k = 1:20 for i = 1:100 if det(floor((2*k + 1)*rand(2) - k)) == 0 percent(k) = percent(k) + 1; end end end percent

a. What do the values of the answer percent tell you about the percent of matrices that are singular as k increases? b. Repeat the experiment for 3 × 3 matrices. What can you say about the percent of singular matrices in this case? c. What does this indicate about the percent of singular matrices as the size of the matrix increases?

Solution

a. Percentage of values with increasing k:

37    22     9     9     4     3     1     2     4     2     1     0     2     2     3     1     2     1     2     0

Therefore, the percentage of matrices decreases exponentially with increase in the value of k.

b. Percentage of values with increasing k for 3X3 matrices:

16     3     5     1     0     0     0     0     1     1     0     1     0     0     0     0     0     0     0     0

For 3*3 matrices, the percentage also decreases exponentially with increase in the value of k.

c. This indicates that the percentage of singular matrices decreases as the size of matrix increases.