91Ó°ÊÓ

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

To create an identity matrix of the order \(m \times m\), use the command \(eye\left( {m,m} \right)\).

To verify\(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\), compute\(\left( {A + I} \right)\left( {A - I} \right) - \left( {{A^2} - I} \right)\).

If the result is 0, then it is true.

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

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

The output of matrix A is shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{0.824168}&{0.893582}&{0.159004}&{0.445958}\\{0.989134}&{0.862390}&{0.286881}&{0.744199}\\{0.336397}&{0.931747}&{0.063368}&{0.620395}\\{0.547681}&{0.558067}&{0.418710}&{0.642377}\end{aligned}} \right)\)

Create a random matrix\(I\)by using the MATLAB command shown below:

\( > > I = eye\left( {4,4} \right)\)

The output of identity matrix I is shown below:

\({\rm{I}} = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{aligned}} \right)\)

Now, run the command\(\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right)\).

\(\begin{aligned}{l}\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right) \sim \left( {\begin{aligned}{*{20}{c}}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{aligned}} \right)\\\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right) \sim {\bf{0}}\end{aligned}\)

It is observed that the difference is the zero matrix.

Hence, \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\) is verified.

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

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

The output of matrix A is shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{0.9658}&{0.6021}&{0.9201}&{0.9057}\\{0.2677}&{0.4002}&{0.9679}&{0.4938}\\{0.3435}&{0.9403}&{0.3527}&{0.6456}\\{0.3338}&{0.1830}&{0.4449}&{0.4819}\end{aligned}} \right)\)

Now, run the command\(\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right)\).

\(\begin{aligned}{l}\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right) \sim \left( {\begin{aligned}{*{20}{c}}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{aligned}} \right)\\\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right) \sim {\bf{0}}\end{aligned}\)

It is observed that the difference is the zero matrix.

Hence, \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\) is verified.

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

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

The output of matrix A is shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{0.956872}&{0.087905}&{0.498262}&{0.205285}\\{0.701973}&{0.963660}&{0.874754}&{0.934861}\\{0.050754}&{0.870641}&{0.594241}&{0.037656}\\{0.066208}&{0.183451}&{0.639346}&{0.603038}\end{aligned}} \right)\)

Now, run the command\(\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right)\).

\(\begin{aligned}{l}\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right) \sim \left( {\begin{aligned}{*{20}{c}}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{aligned}} \right)\\\left( {A + I} \right)*\left( {A - I} \right) - \left( {A*A - I} \right) \sim {\bf{0}}\end{aligned}\)

It is observed that the difference is the zero matrix.

Hence, \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\) is verified.

The equation\(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}\)is true if the MATLAB command is verified, such that\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) = {\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

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

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

The output of matrix A is shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{0.764775}&{0.912129}&{0.556113}&{0.865497}\\{0.712612}&{0.091724}&{0.426475}&{0.869771}\\{0.725032}&{0.260828}&{0.776040}&{0.455232}\\{0.776332}&{0.674258}&{0.884017}&{0.544782}\end{aligned}} \right)\)

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

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

The output of matrix B is shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.287606}&{0.664505}&{0.591512}&{0.415605}\\{0.073937}&{0.797648}&{0.194120}&{0.250417}\\{0.853475}&{0.497764}&{0.698901}&{0.853589}\\{0.890650}&{0.720478}&{0.258672}&{0.832795}\end{aligned}} \right)\)

Now, run the command\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right)\).

\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) = \left( {\begin{aligned}{*{20}{c}}{1.2153}&{ - 1.1835}&{1.2187}&{0.6238}\\{0.7356}&{ - 0.6319}&{0.9272}&{0.3349}\\{0.8986}&{ - 0.5546}&{1.0526}&{0.2155}\\{1.3820}&{ - 0.9062}&{1.2147}&{0.7618}\end{aligned}} \right)\)

Now, run the comment \({\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

\({\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{1.303140}&{0.194832}&{1.190951}&{1.042966}\\{1.125839}&{0.393674}&{1.136243}&{0.759805}\\{0.017409}&{ - 0.732454}&{0.208378}&{ - 0.331142}\\{0.866151}&{ - 0.527435}&{0.824016}&{0.492658}\end{aligned}} \right)\)

It is observed that\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) \ne {\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

Hence, \(\left( {A + B} \right)\left( {A - B} \right) \ne {A^2} - {B^2}\).

The equation\(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}\)is true if the MATLAB command is verified, such that\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) = {\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

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

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

The output of matrix A is shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{0.334451}&{0.130075}&{0.547364}&{0.702031}\\{0.973148}&{0.339488}&{0616595}&{0.011231}\\{0.421252}&{0.319474}&{0.111543}&{0.286304}\\{0.223701}&{0.976674}&{0.372937}&{0.549162}\end{aligned}} \right)\)

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

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

The output of matrix B is shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.6370}&{0.1100}&{0.7787}&{0.6714}\\{0.3517}&{0.3742}&{0.3040}&{0.9532}\\{0.4099}&{0.2457}&{0.8818}&{0.3172}\\{0.7372}&{0.3112}&{0.1004}&{0.9257}\end{aligned}} \right)\)

Now, run the command\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right)\).

\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) = \left( {\begin{aligned}{*{20}{c}}{ - 0.8349}&{1.0230}&{ - 0.7968}&{ - 0.7546}\\{ - 0.4421}&{0.7115}&{ - 0.5297}&{ - 1.0234}\\{ - 0.1989}&{0.4719}&{ - 0.6163}&{ - 0.7649}\\{ - 0.2424}&{0.9909}&{0.2176}&{ - 1.7538}\end{aligned}} \right)\)

And run the command \({\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

\({\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{ - 6.3258e - 01}&{4.3664e - 01}&{ - 6.9744e - 01}&{ - 6.2266e - 01}\\{ - 2.6495e - 01}&{ - 1.0031e - 01}&{6.3469e - 01}&{ - 7.0207e - 01}\\{ - 3.8004e - 01}&{2.6065e - 02}&{ - 6.5658e - 01}&{ - 5.9434e - 01}\\{2.5726e - 03}&{5.0585e - 01}&{1.2092e - 01}&{ - 1.1041e + 00}\end{aligned}} \right)\)

It is observed that\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) \ne {\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

Hence, \(\left( {A + B} \right)\left( {A - B} \right) \ne {A^2} - {B^2}\).

The equation\(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}\)is true if the MATLAB command is verified, such that\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) = {\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

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

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

The output of matrix A is shown below:

\({\rm{A}} = \left( {\begin{aligned}{*{20}{c}}{1.6722e - 01}&{7.9546e - 01}&{4.9693e - 01}&{5.3353e - 03}\\{7.0501e - 01}&{5.3309e - 01}&{8.1323e - 01}&{7.0171e - 01}\\{2.5972e - 01}&{5.9067e - 01}&{8.2506e - 01}&{1.7924 - 01}\\{4.6123e - 01}&{5.4354e - 01}&{7.3495e - 01}&{4.4769e - 01}\end{aligned}} \right)\)

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

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

The output of matrix B is shown below:

\({\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{0.5328}&{0.4403}&{0.6399}&{0.4663}\\{0.3845}&{0.9139}&{0.1529}&{0.8078}\\{0.3646}&{0.6551}&{0.5067}&{0.2168}\\{0.9397}&{0.7910}&{0.3084}&{0.4286}\end{aligned}} \right)\)

Now, run the command\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right)\).

\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) = \left( {\begin{aligned}{*{20}{c}}{ - 0.2047}&{ - 0.4119}&{1.2790}&{ - 0.4875}\\{ - 0.7580}&{ - 0.5999}&{1.7511}&{ - 0.6632}\\{ - 0.1581}&{ - 0.4361}&{1.3262}&{ - 0.4624}\\{ - 0.6131}&{ - 0.2946}&{1.3868}&{ - 0.8098}\end{aligned}} \right)\)

And run the command \({\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

\({\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}} = \left( {\begin{aligned}{*{20}{c}}{ - 0.404306}&{ - 0.571461}&{0.267592}&{ - 0.292110}\\{ - 0.342471}&{ - 0.036869}&{1.258221}&{ - 0.459184}\\{ - 0.077734}&{ - 0.156382}&{0.764773}&{ - 0.257954}\\{ - 0.462302}&{ - 0.343625}&{0.595857}&{ - 0.611723}\end{aligned}} \right)\)

It is observed that\(\left( {{\rm{A}} + {\rm{B}}} \right)*\left( {{\rm{A}} - {\rm{B}}} \right) \ne {\rm{A}}*{\rm{A}} - {\rm{B}}*{\rm{B}}\).

Hence, \(\left( {A + B} \right)\left( {A - B} \right) \ne {A^2} - {B^2}\).