Question

Is it true that \(det{\rm{ }}AB = \left( {det{\rm{ }}A} \right)\left( {det{\rm{ }}B} \right)\)? To find out, generate random \({\bf{5}} \times {\bf{5}}\) matrices A and B, and compute \[det AB - \left( {det A{\rm{ }}det B} \right)\]. Repeat the calculations for three other pairs of \(n \times n\) matrices, for various values of n. Report your results.

Short answer

For various values of \(n\), \(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\) is always true.

Step-by-step solution

Write the MATLAB commands

To create a random\(m \times m\)matrix A, use the MATLAB command

\[ > > A = rand\left( m \right)\].

To compute thedeterminant of matrix A, use the MATLAB command

\( > > det\left( A \right)\).

Find the determinant of the matrix

Use the MATLAB command\(A = rand\left( {\bf{5}} \right)\)to create a random matrix of the order\(5 \times 5\).

\(A = \left[ {\begin{aligned}{*{20}{c}}{0.0782}&{0.7749}&{0.2599}&{0.2638}&{0.5499}\\{0.4427}&{0.8173}&{0.8001}&{0.1455}&{0.1450}\\{0.1067}&{0.8687}&{0.4314}&{0.1361}&{0.8530}\\{0.9619}&{0.0844}&{0.9106}&{0.8693}&{0.6221}\\{0.0046}&{0.3998}&{0.1818}&{0.5797}&{0.3510}\end{aligned}} \right]\)

Use the MATLAB command\(B = rand\left( {\bf{5}} \right)\)to create a random matrix of the order\(5 \times 5\).

\(B = \left[ {\begin{aligned}{*{20}{c}}{0.5132}&{0.1839}&{0.9448}&{0.3692}&{0.4039}\\{0.4018}&{0.2400}&{0.4909}&{0.1112}&{0.0965}\\{0.0760}&{0.4173}&{0.4893}&{0.7803}&{0.1320}\\{0.2399}&{0.0497}&{0.3377}&{0.3897}&{0.9421}\\{0.1233}&{0.9027}&{0.9001}&{0.2417}&{0.9561}\end{aligned}} \right]\)

Obtain the product matrix\(C = AB\)using the MATLAB command\({\bf{C}} = {\bf{A}}*{\bf{B}}\).

\(C = AB = \left[ {\begin{aligned}{*{20}{c}}{0.5023}&{0.8182}&{1.1654}&{0.5535}&{0.9149}\\{0.6692}&{0.7494}&{1.3905}&{0.9704}&{0.6389}\\{0.5744}&{1.1849}&{1.5520}&{0.7318}&{1.1276}\\{0.8821}&{1.1818}&{2.2492}&{1.5642}&{1.9305}\\{0.3592}&{0.5183}&{0.8012}&{0.4988}&{0.9461}\end{aligned}} \right]\)

Compute\[\det AB - \left( {\det A{\rm{ }}\det B} \right)\]using the MATLAB command shown below:

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right)*\det \left( {\rm{B}} \right)\)

So, the output is

\(\begin{aligned}{c}\det {\rm{ }}AB - \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right) = - 5.2042 \times {10^{ - 18}}\\ \approx 0.\end{aligned}\)

Thus,\[\det AB - \left( {\det A{\rm{ }}\det B} \right) = 0\], or\(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Therefore, it is true that \(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Find the determinant of the matrix

Use the MATLAB command\(A = rand\left( {\bf{4}} \right)\)to create a random matrix of the order\(4 \times 4\).

\(A = \left[ {\begin{aligned}{*{20}{c}}{0.5752}&{0.8212}&{0.6491}&{0.5470}\\{0.0598}&{0.0154}&{0.7317}&{0.2963}\\{0.2348}&{0.0430}&{0.6477}&{0.7447}\\{0.3532}&{0.1690}&{0.4509}&{0.1890}\end{aligned}} \right]\)

Use the MATLAB command\(B = rand\left( {\bf{4}} \right)\)to create a random matrix of the order\(4 \times 4\).

\(B = \left[ {\begin{aligned}{*{20}{c}}{0.6868}&{0.7802}&{0.4868}&{0.5085}\\{0.1835}&{0.0811}&{0.4359}&{0.5108}\\{0.3685}&{0.9294}&{0.4468}&{0.8176}\\{0.6256}&{0.7757}&{0.3063}&{0.7948}\end{aligned}} \right]\)

Obtain the product matrix\(C = AB\)using the MATLAB command\({\bf{C}} = {\bf{A}}*{\bf{B}}\).

\(C = AB = \left[ {\begin{aligned}{*{20}{c}}{1.1271}&{1.5430}&{1.0955}&{1.6775}\\{0.4989}&{0.9578}&{0.4535}&{0.8721}\\{0.8737}&{1.3663}&{0.6506}&{1.2629}\\{0.5579}&{0.8549}&{0.5049}&{0.7848}\end{aligned}} \right]\)

Compute\[\det AB - \left( {\det A{\rm{ }}\det B} \right)\]using the MATLAB command shown below:

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right)*\det \left( {\rm{B}} \right)\)

The output is\(\det {\rm{ }}AB - \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right) \approx 0\).

Thus,\[\det AB - \left( {\det A{\rm{ }}\det B} \right) = 0\], or\(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Therefore, it is true that \(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Find the determinant of the matrix

Use the MATLAB command\(A = rand\left( {\bf{3}} \right)\)to create a random matrix of the order\(3 \times 3\).

\(A = \left[ {\begin{aligned}{*{20}{c}}{0.6443}&{0.5328}&{0.8759}\\{0.3786}&{0.3507}&{0.5502}\\{0.8116}&{0.9390}&{0.6225}\end{aligned}} \right]\)

Use the MATLAB command\(B = rand\left( {\bf{3}} \right)\)to create a random matrix of the order\(3 \times 3\).

\(B = \left[ {\begin{aligned}{*{20}{c}}{0.5870}&{0.4709}&{0.1948}\\{0.2077}&{0.2305}&{0.2259}\\{0.3012}&{0.8443}&{0.1707}\end{aligned}} \right]\)

Obtain the product matrix\(C = AB\)using the MATLAB command\({\bf{C}} = {\bf{A}}*{\bf{B}}\).

\(C = AB = \left[ {\begin{aligned}{*{20}{c}}{0.7528}&{1.1658}&{0.3954}\\{0.4609}&{0.7236}&{0.2469}\\{0.8590}&{1.1242}&{0.4765}\end{aligned}} \right]\)

Compute\[\det AB - \left( {\det A{\rm{ }}\det B} \right)\]using MATLAB command as shown below:

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right)*\det \left( {\rm{B}} \right)\)

The output is shown below:

\(\begin{aligned}{c}\det {\rm{ }}AB - \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right) = 8.0231 \times {10^{ - 18}}\\ \approx 0\end{aligned}\)

Thus,\[\det AB - \left( {\det A{\rm{ }}\det B} \right) = 0\], or\(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Therefore, it is true that \(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Find the determinant of the matrix

Use the MATLAB command\(A = rand\left( {\bf{2}} \right)\)to create a random matrix of the order\(2 \times 2\).

\(A = \left[ {\begin{aligned}{*{20}{c}}{0.4609}&{0.2259}\\{0.8443}&{0.6443}\end{aligned}} \right]\)

Use the MATLAB command\(B = rand\left( {\bf{2}} \right)\)to create a random matrix of the order\(2 \times 2\).

\(B = \left[ {\begin{aligned}{*{20}{c}}{1.1271}&{0.7848}\\{1.3663}&{0.6506}\end{aligned}} \right]\)

Obtain the product matrix\(C = AB\)using the MATLAB command\({\bf{C}} = {\bf{A}}*{\bf{B}}\).

\(C = AB = \left[ {\begin{aligned}{*{20}{c}}{0.8281}&{0.5086}\\{1.8319}&{1.0817}\end{aligned}} \right]\)

Compute\[\det AB - \left( {\det A{\rm{ }}\det B} \right)\]using the MATLAB command shown below:

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right)*\det \left( {\rm{B}} \right)\)

The output is\(\det {\rm{ }}AB - \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right) \approx 0\).

Thus,\[\det AB - \left( {\det A{\rm{ }}\det B} \right) = 0\], or\(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).

Therefore, it is true that \(\det {\rm{ }}AB = \left( {\det {\rm{ }}A} \right)\left( {\det {\rm{ }}B} \right)\).