91Ó°ÊÓ

To create a random matrix of order \(m \times m\) with random entries, the command \(rand\left( {m,m} \right)\) is used, and the entries are distributed between 0 and 1.

Test the equality of\({\left( {A + B} \right)^T} = {A^T} + {B^T}\).

Create a random matrix\(A\)by using MATLAB command as shown below:

\( > > A = rand\left( {4,4} \right)\)

The output of matrix A as shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{5.4126e - 01}&{2.8747e - 03}&{5.9284e - 01}&{3.6762e - 01}\\{3.3928e - 01}&{9.2375e - 01}&{5.1778e - 01}&{2.7072e - 01}\\{1.8617e - 01}&{1.3218e - 01}&{5.4036e - 01}&{2.6490e - 01}\\{1.2837e - 01}&{4.2152e - 01}&{2.1465e - 01}&{4.7236e - 01}\end{aligned}} \right)\)

Create a random matrix B by using MATLAB command as shown below:

\( > > B = rand\left( {4,4} \right)\)

The output of matrix B as shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.481133}&{0.344567}&{0.050269}&{0.879742}\\{0.633319}&{0.630218}&{0.605862}&{0.784503}\\{0.927509}&{0.316194}&{0.850075}&{0.044092}\\{0.102503}&{0.850543}&{0.943759}&{0.455563}\end{aligned}} \right)\)

Now, run the command\({\left( {A + B} \right)^\prime }\).

\({\left( {A + B} \right)^\prime } = \left( {\begin{aligned}{*{20}{c}}{1.0224}&{0.9726}&{1.1137}&{0.2309}\\{0.3474}&{1.5540}&{0.4484}&{1.2721}\\{0.6431}&{1.1236}&{1.3904}&{1.1584}\\{1.2474}&{1.0552}&{0.3090}&{1.9279}\end{aligned}} \right)\)

Run the command \(A' + B'\).

\(A' + B' = \left( {\begin{aligned}{*{20}{c}}{1.0224}&{0.9726}&{1.1137}&{0.2309}\\{0.3474}&{1.5540}&{0.4484}&{1.2721}\\{0.6431}&{0.1236}&{1.3904}&{1.1584}\\{1.2474}&{1.0552}&{0.3090}&{0.9279}\end{aligned}} \right)\)

It is observed that\({\left( {A + B} \right)^\prime } = A' + B'\).

Hence, \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) is verified.

Test the equality of\({\left( {A + B} \right)^T} = {A^T} + {B^T}\).

Create a random matrix\(A\)by using MATLAB command as shown below:

\( > > A = rand\left( {4,4} \right)\)

The output of matrix A as shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{5.3126e - 01}&{2.7747e - 03}&{5.9284e - 01}&{3.5762e - 01}\\{3.2928e - 01}&{9.1375e - 01}&{5.0778e - 01}&{2.6072e - 01}\\{1.7617e - 01}&{1.2218e - 01}&{5.3036e - 01}&{2.5490e - 01}\\{1.1837e - 01}&{4.1152e - 01}&{2.0465e - 01}&{4.6236e - 01}\end{aligned}} \right)\)

Create a random matrix B by using MATLAB command as shown below:

\( > > B = rand\left( {4,4} \right)\)

The output of matrix B as shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.381133}&{0.244567}&{0.040269}&{0.779742}\\{0.533319}&{0.530218}&{0.505862}&{0.684503}\\{0.827509}&{0.216194}&{0.750075}&{0.034092}\\{0.002503}&{0.750543}&{0.843759}&{0.355563}\end{aligned}} \right)\)

Now, run the command \({\left( {A + B} \right)^\prime }\).

Now, run the command\({\left( {A + B} \right)^\prime }\).

\({\left( {A + B} \right)^\prime } = \left( {\begin{aligned}{*{20}{c}}{1.0124}&{0.8726}&{1.0137}&{0.1309}\\{0.2474}&{1.4540}&{0.3484}&{1.1721}\\{0.5431}&{1.0236}&{1.2904}&{1.0584}\\{1.1474}&{1.0452}&{0.2090}&{1.8279}\end{aligned}} \right)\)

Run the command \(A' + B'\).

\(A' + B' = \left( {\begin{aligned}{*{20}{c}}{1.0124}&{0.8726}&{1.0137}&{0.1309}\\{0.2474}&{1.4540}&{0.3484}&{1.1721}\\{0.5431}&{0.0236}&{1.2904}&{1.0584}\\{1.1474}&{1.0452}&{0.2090}&{0.8279}\end{aligned}} \right)\)

It is observed that\({\left( {A + B} \right)^\prime } = A' + B'\).

Hence, \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) is verified.

Test the equality of\({\left( {A + B} \right)^T} = {A^T} + {B^T}\).

Create a random matrix\(A\)by using MATLAB command as shown below:

\( > > A = rand\left( {4,4} \right)\)

The output of matrix A as shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{5.2126e - 01}&{2.6747e - 03}&{5.9284e - 01}&{3.4762e - 01}\\{3.1928e - 01}&{9.0375e - 01}&{5.0678e - 01}&{2.5072e - 01}\\{1.6617e - 01}&{1.1218e - 01}&{5.2036e - 01}&{2.4490e - 01}\\{1.0837e - 01}&{4.0152e - 01}&{2.0365e - 01}&{4.5236e - 01}\end{aligned}} \right)\)

Create a random matrix B by using MATLAB command as shown below:

\( > > B = rand\left( {4,4} \right)\)

The output of matrix B as shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.281133}&{0.144567}&{0.030269}&{0.679742}\\{0.433319}&{0.430218}&{0.405862}&{0.584503}\\{0.727509}&{0.116194}&{0.650075}&{0.024092}\\{0.001503}&{0.650543}&{0.743759}&{0.255563}\end{aligned}} \right)\)

Now, run the command\({\left( {A + B} \right)^\prime }\).

\({\left( {A + B} \right)^\prime } = \left( {\begin{aligned}{*{20}{c}}{1.0024}&{0.7726}&{1.0037}&{0.0309}\\{0.1474}&{1.3540}&{0.2484}&{1.0721}\\{0.4431}&{1.0136}&{1.1904}&{1.0484}\\{1.0474}&{1.0352}&{0.1090}&{1.7279}\end{aligned}} \right)\)

Run the command \(A' + B'\).

\(A' + B' = \left( {\begin{aligned}{*{20}{c}}{1.0024}&{0.7726}&{1.0037}&{0.0309}\\{0.1474}&{1.3540}&{0.2484}&{1.0721}\\{0.4431}&{0.0136}&{1.1904}&{1.0484}\\{1.0474}&{1.0352}&{0.1090}&{0.7279}\end{aligned}} \right)\)

It is observed that\({\left( {A + B} \right)^\prime } = A' + B'\).

Hence, \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) is verified.

Test the equality of\({\left( {AB} \right)^T} = {A^T}{B^T}\).

Create a random matrix\(A\)by using MATLAB command as shown below:

\( > > A = rand\left( {4,4} \right)\)

The output of matrix A as shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{5.2126e - 01}&{2.6747e - 03}&{5.9284e - 01}&{3.4762e - 01}\\{3.1928e - 01}&{9.0375e - 01}&{5.0678e - 01}&{2.5072e - 01}\\{1.6617e - 01}&{1.1218e - 01}&{5.2036e - 01}&{2.4490e - 01}\\{1.0837e - 01}&{4.0152e - 01}&{2.0365e - 01}&{4.5236e - 01}\end{aligned}} \right)\)

Create a random matrix B by using MATLAB command as shown below:

\( > > B = rand\left( {4,4} \right)\)

The output of matrix B as shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.281133}&{0.144567}&{0.030269}&{0.679742}\\{0.433319}&{0.430218}&{0.405862}&{0.584503}\\{0.727509}&{0.116194}&{0.650075}&{0.024092}\\{0.001503}&{0.650543}&{0.743759}&{0.255563}\end{aligned}} \right)\)

Now, run the command\({\left( {A*B} \right)^\prime }\).

\({\left( {A*B} \right)^\prime } = \left( {\begin{aligned}{*{20}{c}}{0.8498}&{1.2563}&{0.7016}&{0.5762}\\{0.6884}&{1.0930}&{0.5436}&{0.7795}\\{0.8799}&{1.2724}&{0.7988}&{0.8901}\\{0.6720}&{1.1693}&{0.4120}&{0.6683}\end{aligned}} \right)\)

Run the command \(A'*B'\).

\(A'*B' = \left( {\begin{aligned}{*{20}{c}}{1.0024}&{0.7726}&{1.0037}&{0.0309}\\{0.1474}&{1.3540}&{0.2484}&{1.0721}\\{0.4431}&{0.0136}&{1.1904}&{1.0484}\\{1.0474}&{1.0352}&{0.1090}&{0.7279}\end{aligned}} \right)\)

It is observed that\({\left( {A*B} \right)^\prime } \ne A'*B'\).

Hence, the equality \({\left( {AB} \right)^T} = {A^T}{B^T}\) is incorrect.

Test the equality of\({\left( {AB} \right)^T} = {A^T}{B^T}\).

Create a random matrix\(A\)by using MATLAB command as shown below:

\( > > A = rand\left( {4,4} \right)\)

The output of matrix A as shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{5.3126e - 01}&{2.7747e - 03}&{5.9284e - 01}&{3.5762e - 01}\\{3.2928e - 01}&{9.1375e - 01}&{5.0778e - 01}&{2.6072e - 01}\\{1.7617e - 01}&{1.2218e - 01}&{5.3036e - 01}&{2.5490e - 01}\\{1.1837e - 01}&{4.1152e - 01}&{2.0465e - 01}&{4.6236e - 01}\end{aligned}} \right)\)

Create a random matrix B by using MATLAB command as shown below:

\( > > B = rand\left( {4,4} \right)\)

The output of matrix B as shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.381133}&{0.244567}&{0.040269}&{0.779742}\\{0.533319}&{0.530218}&{0.505862}&{0.684503}\\{0.827509}&{0.216194}&{0.750075}&{0.034092}\\{0.002503}&{0.750543}&{0.843759}&{0.355563}\end{aligned}} \right)\)

Now, run the command\({\left( {A*B} \right)^\prime }\).

\({\left( {A*B} \right)^\prime } = \left( {\begin{aligned}{*{20}{c}}{0.7498}&{1.1563}&{0.6016}&{0.4762}\\{0.5884}&{1.0830}&{0.4436}&{0.6795}\\{0.7799}&{1.1724}&{0.6988}&{0.7901}\\{0.5720}&{1.0693}&{0.3120}&{0.5683}\end{aligned}} \right)\)

Run the command \(A'*B'\).

\(A'*B' = \left( {\begin{aligned}{*{20}{c}}{1.0014}&{0.6726}&{1.0027}&{0.0209}\\{0.0474}&{1.2540}&{0.1484}&{1.0621}\\{0.3431}&{0.0036}&{1.0904}&{1.0384}\\{1.0374}&{1.0252}&{0.0090}&{0.6279}\end{aligned}} \right)\)

It is observed that\({\left( {A*B} \right)^\prime } \ne A'*B'\).

Hence, the equality \({\left( {AB} \right)^T} = {A^T}{B^T}\) is incorrect.

Create a random matrix\(A\)by using MATLAB command as shown below:

\( > > A = rand\left( {4,4} \right)\)

The output of matrix A as shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{5.2126e - 01}&{2.6747e - 03}&{5.9284e - 01}&{3.4762e - 01}\\{3.1928e - 01}&{9.0375e - 01}&{5.0678e - 01}&{2.5072e - 01}\\{1.6617e - 01}&{1.1218e - 01}&{5.2036e - 01}&{2.4490e - 01}\\{1.0837e - 01}&{4.0152e - 01}&{2.0365e - 01}&{4.5236e - 01}\end{aligned}} \right)\)

Create a random matrix B by using MATLAB command as shown below:

\( > > B = rand\left( {4,4} \right)\)

The output of matrix B as shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.281133}&{0.144567}&{0.030269}&{0.679742}\\{0.433319}&{0.430218}&{0.405862}&{0.584503}\\{0.727509}&{0.116194}&{0.650075}&{0.024092}\\{0.001503}&{0.650543}&{0.743759}&{0.255563}\end{aligned}} \right)\)

Now, run the command\({\left( {A*B} \right)^\prime }\).

\({\left( {A*B} \right)^\prime } = \left( {\begin{aligned}{*{20}{c}}{0.6498}&{1.0563}&{0.5016}&{0.3762}\\{0.4884}&{1.0730}&{0.3436}&{0.5795}\\{0.6799}&{1.0724}&{0.5988}&{0.6901}\\{0.4720}&{1.0593}&{0.2120}&{0.4683}\end{aligned}} \right)\)

Run the command \(A'*B'\).

\(A'*B' = \left( {\begin{aligned}{*{20}{c}}{1.0004}&{0.5726}&{1.0017}&{0.0109}\\{0.0374}&{1.1540}&{0.0484}&{1.0521}\\{0.2431}&{0.0026}&{1.0804}&{1.0284}\\{1.0274}&{1.0152}&{0.0080}&{0.5279}\end{aligned}} \right)\)

It is observed that\({\left( {A*B} \right)^\prime } \ne A'*B'\).

Hence, the equality\({\left( {AB} \right)^T} = {A^T}{B^T}\)is incorrect.