Question

Construct a random \({\bf{4}} \times {\bf{4}}\) matrix A with integer entries between \( - {\bf{9}}\) and 9, and compare det A with det\({A^T}\), \(det\left( { - A} \right)\), \(det\left( {{\bf{2}}A} \right)\), and \(det\left( {{\bf{10}}A} \right)\). Repeat with two other random \({\bf{4}} \times {\bf{4}}\) integer matrices, and make conjectures about how these determinants are related. (Refer to Exercise 36 in Section 2.1.) Then check your conjectures with several random \({\bf{5}} \times {\bf{5}}\) and \({\bf{6}} \times {\bf{6}}\) integer matrices. Modify your conjectures, if necessary, and report your results.

Short answer

For an\(n \times n\)matrix,\(\det \left( {kA} \right) = {k^n}\det \left( A \right)\), where k is an integer.

Step-by-step solution

Write the MATLAB commands

To create a random\(m \times m\)matrix A with integer entries between\( - a\)and\(a\), the MATLAB command is

\( > > A = randi\left( {\left( { - a,a} \right),m,m} \right)\).

To compute thedeterminant of matrix A, the MATLAB command is

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

Find the determinant of the matrix

Use the MATLAB command\(A = randi\left( {\left( { - {\bf{9}},9} \right),4,4} \right)\)to create a random matrix of the order\(4 \times 4\)between the integers\( - 9\)and 9.

\(A = \left( {\begin{aligned}{*{20}{c}}2&4&2&3\\5&5&9&6\\5&6&4&1\\2&4&0&6\end{aligned}} \right)\)

Compute the determinant of matrix A by using the MATLAB command shown below:

\( > > {\rm{d}} = \det \left( {\rm{A}} \right)\)

The output obtained is\({\rm{d}} = 232\).

Therefore, the determinant is\(\det \left( A \right) = 232\).

The transpose of the matrix is shown below:

\({A^T} = \left( {\begin{aligned}{*{20}{c}}2&5&5&2\\4&5&6&4\\2&9&4&0\\3&6&1&6\end{aligned}} \right)\)

Obtain the determinant of the transpose of matrix A as shown below:

\( > > {\rm{d}} = \det \left( {{{\rm{A}}^T}} \right)\)

So,\(\det \left( {{{\rm{A}}^T}} \right) = 232\).

Obtain the\( - A\)matrix as shown below:

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

Obtain the determinant of the matrix\( - A\)as shown below:

\( > > {\rm{d}} = \det \left( { - {\rm{A}}} \right)\)

So,\(\det \left( { - {\rm{A}}} \right) = 232\).

Obtain the\(2A\)matrix as shown below:

\(2A = \left( {\begin{aligned}{*{20}{c}}4&8&4&6\\{10}&{10}&{18}&{12}\\{10}&{12}&8&2\\4&8&0&{12}\end{aligned}} \right)\)

Obtain the determinant of the matrix\(2A\)as shown below:

\( > > {\rm{d}} = \det \left( {{\rm{2A}}} \right)\)

So,\(\det \left( {{\rm{2A}}} \right) = 3712 = {2^4} \times 232\).

Obtain the\(10A\)matrix as shown below:

\(10A = \left( {\begin{aligned}{*{20}{c}}{20}&{40}&{20}&{30}\\{50}&{50}&{90}&{60}\\{50}&{60}&{40}&{10}\\{20}&{40}&0&{60}\end{aligned}} \right)\)

Obtain the determinant of the matrix\(10A\)as shown below:

\( > > {\rm{d}} = \det \left( {{\rm{10A}}} \right)\)

So,\(\det \left( {{\rm{2A}}} \right) = 2320000 = {10^4} \times 232\).

Thus, for an\(n \times n\)matrix,\(\det \left( {kA} \right) = {k^n}\det \left( A \right)\), where k is an integer.

Find the determinant of the matrix

Use the MATLAB command\(A = randi\left( {\left( { - {\bf{9}},9} \right),4,4} \right)\)to create a random matrix of the order\(4 \times 4\), between the integers\( - 9\)and 9.

\(A = \left( {\begin{aligned}{*{20}{c}}3&0&1&2\\6&2&4&1\\2&7&5&5\\0&1&2&3\end{aligned}} \right)\)

Compute the determinant of matrix A using the MATLAB command shown below:

\( > > {\rm{d}} = \det \left( {\rm{A}} \right)\)

Thus, the output obtained is\({\rm{d}} = - 157\).

Therefore, the determinant is\(\det \left( A \right) = - 157\).

The transpose of the matrix is

\({A^T} = \left( {\begin{aligned}{*{20}{c}}3&6&2&0\\0&2&7&1\\1&4&5&2\\2&1&5&3\end{aligned}} \right)\).

Obtain the determinant of the transpose of matrix A as shown below:

\( > > {\rm{d}} = \det \left( {{{\rm{A}}^T}} \right)\)

So,\(\det \left( {{{\rm{A}}^T}} \right) = - 157\).

Obtain the\( - A\)matrix as shown below:

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

Obtain the determinant of matrix\( - A\)as shown below:

\( > > {\rm{d}} = \det \left( { - {\rm{A}}} \right)\)

So,\(\det \left( { - {\rm{A}}} \right) = - 157\).

Obtain the\(2A\)matrix as shown below:

\(2A = \left( {\begin{aligned}{*{20}{c}}6&0&2&4\\{12}&4&8&2\\4&{14}&{10}&{10}\\0&2&4&6\end{aligned}} \right)\)

Obtain the determinant of the matrix\(2A\)as shown below:

\( > > {\rm{d}} = \det \left( {{\rm{2A}}} \right)\)

So,\(\det \left( {{\rm{2A}}} \right) = - 2512 = {2^4} \times \left( { - 157} \right)\).

Obtain the\(10A\)matrix as shown below:

\(10A = \left( {\begin{aligned}{*{20}{c}}{30}&0&{10}&{20}\\{60}&{20}&{40}&{10}\\{20}&{70}&{50}&{50}\\0&{10}&{20}&{30}\end{aligned}} \right)\)

Obtain the determinant of the matrix\(10A\)as shown below:

\( > > {\rm{d}} = \det \left( {{\rm{10A}}} \right)\)

So,\(\det \left( {{\rm{2A}}} \right) = - 1570000 = {10^4} \times \left( { - 157} \right)\).

Thus, for an\(n \times n\)matrix\(\det \left( {kA} \right) = {k^n}\det \left( A \right)\), where k is an integer.

Find the determinant of the matrix

Use the MATLAB command\(A = randi\left( {\left( { - {\bf{9}},9} \right),4,4} \right)\)to create a random matrix of order\(4 \times 4\), between the integers\( - 9\)and 9.

\(A = \left( {\begin{aligned}{*{20}{c}}2&4&6&0\\1&1&0&3\\5&6&2&1\\2&5&6&0\end{aligned}} \right)\)

Compute the determinant of matrix A using the MATLAB command shown below:

\( > > {\rm{d}} = \det \left( {\rm{A}} \right)\)

Thus, the output obtained is\({\rm{d}} = 72\).

And the determinant is\(\det \left( A \right) = 72\).

The transpose of the matrix is

\({A^T} = \left( {\begin{aligned}{*{20}{c}}2&1&5&2\\4&1&6&5\\6&0&2&6\\0&3&1&0\end{aligned}} \right)\).

Obtain the determinant of the transpose of matrix A as shown below:

\( > > {\rm{d}} = \det \left( {{{\rm{A}}^T}} \right)\)

So,\(\det \left( {{{\rm{A}}^T}} \right) = 72\).

Obtain the\( - A\)matrix as shown below:

\( - A = \left( {\begin{aligned}{*{20}{c}}{ - 2}&{ - 4}&{ - 6}&0\\{ - 1}&{ - 1}&0&{ - 3}\\{ - 5}&{ - 6}&{ - 2}&{ - 1}\\{ - 2}&{ - 5}&{ - 6}&0\end{aligned}} \right)\)

Obtain the determinant of matrix\( - A\)as shown below:

\( > > {\rm{d}} = \det \left( { - {\rm{A}}} \right)\)

So,\(\det \left( { - {\rm{A}}} \right) = 72\).

Obtain the\(2A\)matrix as shown below:

\(2A = \left( {\begin{aligned}{*{20}{c}}4&8&{12}&0\\2&2&0&6\\{10}&{12}&4&2\\4&{10}&{12}&0\end{aligned}} \right)\)

Obtain the determinant of the matrix\(2A\)as shown below:

\( > > {\rm{d}} = \det \left( {{\rm{2A}}} \right)\)

So,\(\det \left( {{\rm{2A}}} \right) = 1152 = {2^4} \times 72\).

Obtain the\(10A\)matrix as shown below:

\(10A = \left( {\begin{aligned}{*{20}{c}}{20}&{40}&{60}&0\\{10}&{10}&0&{30}\\{50}&{60}&{20}&{10}\\{20}&{50}&{60}&0\end{aligned}} \right)\)

Obtain the determinant of the matrix\(10A\)as shown below:

\( > > {\rm{d}} = \det \left( {{\rm{10A}}} \right)\)

So,\(\det \left( {{\rm{2A}}} \right) = 720000 = {10^4} \times 72\).

Thus, for an\(n \times n\)matrix,\(\det \left( {kA} \right) = {k^n}\det \left( A \right)\), where k is an integer.