Question

Exercises 31 and 32 should be solved without performing row operations. [Hint: Write \(Ax = 0\) as a vector equation.]

32. Given \(A = \left[ {\begin{array}{*{20}{c}}4&1&6\\{ - 7}&5&3\\9&{ - 3}&3\end{array}} \right]\) . Observe that the first column plus twice the second column equals the third column. Find a nontrivial solution of \(Ax = 0\).

Short answer

\(Ax = 0\) is a matrix equation for \(x = \left( {1,2, - 1} \right)\).

Step-by-step solution

Write the matrix as an expression

Write matrix\(A\)as the expression\(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right]\).

The sum of the first two columns equals the third column, which indicates that \({a_1} + 2{a_2} = {a_3}\).

Rewrite the given equation

Rewrite the equation\({a_1} + 2{a_2} = {a_3}\)as\({a_1} + 2{a_2} - {a_3} = 0\).

Thus, \(Ax = 0\) is a matrix equation for \(x = \left( {1,2, - 1} \right)\).