Question

Question 46: (M) If \(\det A\) is close to zero, is the matrix \(A\) nearly singular? Experiment with the nearly singular \(4 \times 4\) matrix

\(A = \left( {\begin{array}{*{20}{c}}4&0&{ - 7}&{ - 7}\\{ - 6}&1&{11}&9\\7&{ - 5}&{10}&{19}\\{ - 1}&2&3&{ - 1}\end{array}} \right)\)

Compute the determinants of \(A,10A,\)and\(0.1A\). In contrast, compute the condition numbers of these matrices. Repeat these calculation when \(A\) is the \(4 \times 4\) identity matrix. Discuss your results.

Short answer

The determinants of \(A,10A\) and \(0.1A\) are \(1,\)\(10,000\), and \(0.0001\).

The condition number of matrices\(A,10A\) and \(0.1A\)is \(23683\).

Step-by-step solution

Compute the determinant of \(A,10A\) and \(0.1A\)

The \(\det A\) for this matrix is calculated as 1 and cond\(A \approx 23683\). Matrix \(A\) is not invertible because \(\det A \ne 0\).

Use the following MATLAB code to compute the inverse of matrix \(A\):

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {4\,\,\,0\,\,\, - 7\,\,\, - 7;\,\, - 6\,\,\,1\,\,\,11\,\,\,9;\,\,7\,\,\, - 5\,\,\,10\,\,\,19;\, - 1\,\,\,2\,\,\,3\,\,\, - 1} \right)\\ > > {\mathop{\rm inv}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

\({{\mathop{\rm A}\nolimits} ^{ - 1}} = \left( {\begin{array}{*{20}{c}}{ - 19}&{ - 14}&0&7\\{ - 549}&{ - 401}&{ - 2}&{196}\\{267}&{195}&1&{ - 95}\\{ - 278}&{ - 203}&{ - 1}&{99}\end{array}} \right)\)

The determinant is extremely sensitive to scale as \(\det 10A = {10^4}\det A = 10,000\), and \(\det 0.1A = {\left( {0.1} \right)^4}\det A = 0.0001\).

Thus, the determinants of \(A,10A\) and \(0.1A\)are \(1,\)\(10,000\), and \(0.0001\).

Compute the condition number of these matrices

Scaling does not change the condition number: \({\mathop{\rm cond}\nolimits} \left( {10A} \right) = {\mathop{\rm cond}\nolimits} \left( {0.1A} \right) = {\mathop{\rm cond}\nolimits} A\)\( \approx 23683\).

Thus, the condition number of matrices\(A,10A\) and \(0.1A\)is \(23683\).

Step 3:Establish \(A\) as the identity matrix

If \(A = {I_4}\), then \(\det A = 1\) and \({\mathop{\rm cond}\nolimits} A = 1\). The determinant is also sensitive to scaling as before the values were\(\det 10A = {10^4}\det A = 10,000\)and \(\det 0.1A = {\left( {0.1} \right)^4}\det A = 0.001\). However, scaling does not change the condition number: \({\mathop{\rm cond}\nolimits} \left( {10A} \right) = {\mathop{\rm cond}\nolimits} \left( {0.1A} \right) = {\mathop{\rm cond}\nolimits} A\)\( = 1\).