Question

Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right]\) is a solution of \(Ax = 0\).

Short answer

Construct \(A\) so that the sum of the first and third columns is twice the sum of the second column.

Step-by-step solution

Write the \(3 \times 3\) non-zero matrix \(A\) 

Consider the \(3 \times 3\) non-zero matrix \(A\) is \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right]\).

Use the equation \(Ax = 0\) to construct matrix \(A\) 

Write the matrix as \(Ax = 0\).

\(\begin{array}{l}\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\1\end{array}} \right] = 0\\1 \cdot {a_1} - 2 \cdot {a_2} + 1 \cdot {a_1} = 0\end{array}\)

Thus, construct matrix \(A\) in such a way that the sum of the first and third columns is twice the sum of the second column.