Question

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

31. Given \(A = \left[ {\begin{array}{*{20}{c}}2&3&5\\{ - 5}&1&{ - 4}\\{ - 3}&{ - 1}&{ - 4}\\1&0&1\end{array}} \right]\) . Observe that the third column is the sum of the first two columns. Find a nontrivial solution of \(Ax = 0\).

Short answer

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

Step-by-step solution

Write the matrix as an expression

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

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

Rewrite the given equation

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

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