Question

Construct a \(3 \times 3\) nonzero matrix Asuch that the vector \(\left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right]\) is a solution of \(A{\bf{x}} = 0\).

Short answer

The solution set in matrix form is \(A = \left[ {\begin{array}{*{20}{c}}{ - {a_2} - {a_3}}&{{a_2}}&{{a_3}}\\{ - {b_2} - {b_3}}&{{b_2}}&{{b_3}}\\{ - {c_2} - {c_3}}&{{c_2}}&{{c_3}}\end{array}} \right]\).

Step-by-step solution

Construct a \(3 \times 3\) nonzero matrix A

Since the order of matrix A is \(3 \times 3\), so matrix A consists of 3 rows and 3 columns as shown below:

Let \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\).

Write the matrix equation \(A{\bf{x}} = 0\)

Consider matrix \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\), and the given vector \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right]\).

In the matrix equation form, it can be written as shown below:

\(\begin{array}{c}A{\bf{x}} = 0\\\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{array}\)

Solve the matrix equation \(A{\bf{x}} = 0\)

\(\begin{array}{c}1 \cdot \left[ {\begin{array}{*{20}{c}}{{a_1}}\\{{b_1}}\\{{c_1}}\end{array}} \right] + 1 \cdot \left[ {\begin{array}{*{20}{c}}{{a_2}}\\{{b_2}}\\{{c_2}}\end{array}} \right] + 1 \cdot \left[ {\begin{array}{*{20}{c}}{{a_3}}\\{{b_3}}\\{{c_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}{{a_1} + {a_2} + {a_3}}\\{{b_1} + {b_2} + {b_3}}\\{{c_1} + {c_2} + {c_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{array}\)

Thus, the solution set is shown below:

\(\begin{array}{l}{a_1} + {a_2} + {a_3} = 0\\{b_1} + {b_2} + {b_3} = 0\\{c_1} + {c_2} + {c_3} = 0\end{array}\)

Or, it can also be represented as shown below:

\[\begin{array}{l}{a_1} = - {a_2} - {a_3}\\{b_1} = - {b_2} - {b_3}\\{c_1} = - {c_2} - {c_3}\end{array}\]

Construct the matrix in the solution set form

The matrix can be represented by using \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]\) as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{ - {a_2} - {a_3}}&{{a_2}}&{{a_3}}\\{ - {b_2} - {b_3}}&{{b_2}}&{{b_3}}\\{ - {c_2} - {c_3}}&{{c_2}}&{{c_3}}\end{array}} \right]\)

Thus, the solution set in matrix form is \(A = \left[ {\begin{array}{*{20}{c}}{ - {a_2} - {a_3}}&{{a_2}}&{{a_3}}\\{ - {b_2} - {b_3}}&{{b_2}}&{{b_3}}\\{ - {c_2} - {c_3}}&{{c_2}}&{{c_3}}\end{array}} \right]\) and it satisfies \(A{\bf{x}} = 0\).