Question

Is it true that \(det \left( {A + B} \right) = det A + det B\)? Experiment with four pairs of random matrices as in Exercise 44, and make a conjecture.

Short answer

For various values of \(n\), \(\det \left( {A + B} \right) = \det A + \det B\) is not 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 order\(5 \times 5\), as shown below:

\(A = \left( {\begin{aligned}{*{20}{c}}{0.2277}&{0.1848}&{0.2581}&{0.7112}&{0.4242}\\{0.4357}&{0.9049}&{0.4087}&{0.2217}&{0.5079}\\{0.3111}&{0.9797}&{0.5649}&{0.1174}&{0.0855}\\{0.9234}&{0.4389}&{0.2622}&{0.2967}&{0.2625}\\{0.4302}&{0.1111}&{0.6028}&{0.3188}&{0.8010}\end{aligned}} \right)\)

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

\(B = \left( {\begin{aligned}{*{20}{c}}{0.0292}&{0.2373}&{0.2316}&{0.3674}&{0.7962}\\{0.9289}&{0.4588}&{0.4889}&{0.9880}&{0.0987}\\{0.7302}&{0.9631}&{0.6241}&{0.0371}&{0.2619}\\{0.4886}&{0.5468}&{0.6791}&{0.8852}&{0.3354}\\{0.5785}&{0.5211}&{0.3955}&{0.9133}&{0.6797}\end{aligned}} \right)\)

Obtain the matrix\(C = A + B\)using the MATLAB command\({\bf{C}} = {\bf{A}} + {\bf{B}}\), as shown below:

\(C = A + B = \left( {\begin{aligned}{*{20}{c}}{0.2569}&{0.4221}&{0.4897}&{1.0787}&{1.2204}\\{1.3646}&{0.3637}&{0.8976}&{1.2097}&{0.6066}\\{1.0414}&{1.9428}&{1.2190}&{0.1552}&{0.3474}\\{1.4120}&{0.9857}&{0.9413}&{1.1818}&{0.5978}\\{1.0087}&{0.6323}&{0.9984}&{1.2321}&{1.4807}\end{aligned}} \right)\)

Compute\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right)\)using the MATLAB command shown below:

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

The output is\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right) - \det \left( {\rm{B}} \right) = - 0.0594\).

Since the difference is not zero,\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right) \ne 0\), or\(\det \left( {A + B} \right) \ne \det A + \det B\).

Therefore, it is not always true that \(\det \left( {A + B} \right) \ne \det A + \det B\).

Find the determinant of the matrix

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

\(A = \left( {\begin{aligned}{*{20}{c}}{0.1366}&{0.4942}&{0.8909}&{0.0305}\\{0.7212}&{0.7791}&{0.3342}&{0.7441}\\{0.1068}&{0.7150}&{0.6987}&{0.5000}\\{0.6538}&{0.9037}&{0.1978}&{0.4799}\end{aligned}} \right)\)

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

\(B = \left( {\begin{aligned}{*{20}{c}}{0.9047}&{0.8055}&{0.8865}&{0.9787}\\{0.6099}&{0.5767}&{0.0287}&{0.7127}\\{0.6177}&{0.1829}&{0.4899}&{0.5005}\\{0.8594}&{0.2399}&{0.1679}&{0.4711}\end{aligned}} \right)\)

Obtain the matrix\(C = A + B\)using the MATLAB command\({\bf{C}} = {\bf{A}} + {\bf{B}}\), as shown below:

\(C = A + B = \left( {\begin{aligned}{*{20}{c}}{1.0413}&{1.2997}&{1.7774}&{1.0092}\\{0.3311}&{0.3558}&{0.3628}&{1.4568}\\{0.7244}&{0.8980}&{1.1886}&{1.0005}\\{1.5132}&{1.1437}&{0.3657}&{0.9510}\end{aligned}} \right)\)

Compute\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right)\)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

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right) - \det \left( {\rm{B}} \right) = 0.2872\).

Since the difference is not zero,\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right) \ne 0\), or\(\det \left( {A + B} \right) \ne \det A + \det B\).

Therefore, it is not always true that \(\det \left( {A + B} \right) \ne \det A + \det B\).

Find the determinant of the matrix

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

\(A = \left( {\begin{aligned}{*{20}{c}}{0.0596}&{0.0714}&{0.8181}\\{0.6820}&{0.5216}&{0.8175}\\{0.0424}&{0.0967}&{0.7224}\end{aligned}} \right)\)

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

\(B = \left( {\begin{aligned}{*{20}{c}}{0.1499}&{0.9730}&{0.4538}\\{0.6596}&{0.6490}&{0.4324}\\{0.5186}&{0.8003}&{0.8253}\end{aligned}} \right)\)

Obtain the matrix\(C = A + B\)using the MATLAB command\({\bf{C}} = {\bf{A}} + {\bf{B}}\)as shown below:

\(C = A + B = \left( {\begin{aligned}{*{20}{c}}{0.2095}&{1.0444}&{1.2719}\\{1.3416}&{1.1706}&{1.2499}\\{0.5610}&{0.8971}&{1.5478}\end{aligned}} \right)\)

Compute\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right)\)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 shown below:

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right) - \det \left( {\rm{B}} \right) = - 0.4208\)

Since the difference is not zero,\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right) \ne 0\), or\(\det \left( {A + B} \right) \ne \det A + \det B\).

Therefore, it is not always true that \(\det \left( {A + B} \right) \ne \det A + \det B\).

Determinant of the matrix

Use the MATLAB command\(A = rand\left( {\bf{6}} \right)\)to create a random matrix of the order\(6 \times 6\).

\(A = \left( {\begin{aligned}{*{20}{c}}{0.0835}&{0.0605}&{0.2920}&{0.3724}&{0.0527}&{0.4177}\\{0.1332}&{0.3993}&{0.4317}&{0.1981}&{0.7379}&{0.9831}\\{0.1734}&{0.5269}&{0.0155}&{0.4897}&{0.2691}&{0.3015}\\{0.3909}&{0.4168}&{0.9841}&{0.3395}&{0.4228}&{0.7011}\\{0.8314}&{0.6569}&{0.1672}&{0.9516}&{0.5479}&{0.6663}\\{0.8034}&{0.6280}&{0.1062}&{0.9203}&{0.9427}&{0.5391}\end{aligned}} \right)\)

Use the MATLAB command\(B = rand\left( {\bf{6}} \right)\)to create a random matrix of the order\(6 \times 6\).

\(B = \left( {\begin{aligned}{*{20}{c}}{0.6981}&{0.0326}&{0.4607}&{0.1909}&{0.3846}&{0.8244}\\{0.6665}&{0.5612}&{0.9816}&{0.4283}&{0.5830}&{0.9827}\\{0.1781}&{0.8819}&{0.1564}&{0.4820}&{0.2518}&{0.7302}\\{0.1280}&{0.6692}&{0.8555}&{0.1206}&{0.2904}&{0.3439}\\{0.9991}&{0.1904}&{0.6448}&{0.5895}&{0.6171}&{0.5841}\\{0.1711}&{0.3689}&{0.3763}&{0.2262}&{0.2653}&{0.1078}\end{aligned}} \right)\)

Obtain the matrix\(C = A + B\)using the MATLAB command\({\bf{C}} = {\bf{A}} + {\bf{B}}\)shown below:

\(C = A + B = \left( {\begin{aligned}{*{20}{c}}{0.7816}&{0.0931}&{0.7527}&{0.5633}&{0.4373}&{1.2421}\\{0.7997}&{0.9605}&{1.4133}&{0.6264}&{1.3208}&{1.9657}\\{0.3515}&{1.4087}&{0.1719}&{0.9717}&{0.5209}&{1.0317}\\{0.5190}&{1.0860}&{1.8396}&{0.4601}&{0.7133}&{1.0450}\\{1.8305}&{0.8473}&{0.8119}&{1.5411}&{1.1650}&{1.2504}\\{0.9745}&{0.9969}&{0.4825}&{1.1465}&{1.2080}&{1.6469}\end{aligned}} \right)\)

Compute\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right)\)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 shown below:

\( > > \det \left( {\rm{C}} \right) - \det \left( {\rm{A}} \right) - \det \left( {\rm{B}} \right) = - 0.5297\)

Since the difference is not zero,\(\det \left( {A + B} \right) - \left( {\det A + \det B} \right) \ne 0\)or\(\det \left( {A + B} \right) \ne \det A + \det B\).

Therefore, it is not always true that \(\det \left( {A + B} \right) \ne \det A + \det B\).