Question

Let \(A = \left( {\begin{array}{*{20}{c}}2&{ - 1}\\{ - 6}&3\end{array}} \right)\) and \(b = \left( {\begin{array}{*{20}{c}}{{b_1}}\\{{b_2}}\end{array}} \right)\). Show that the equation \(Ax = {\mathop{\rm b}\nolimits} \) does not have a solution for all possible \({\mathop{\rm b}\nolimits} \), and describe the set of all b for which \(Ax = {\mathop{\rm b}\nolimits} \) does have a solution.

Short answer

The equation \(Ax = b\) is inconsistent when \(3{b_1} + {b_2}\) is non-zero. The set b for which the equation \(Ax = b\) is consistent can be a line passing through the origin for all the set of points \(\left( {{b_1},{b_2}} \right)\) satisfying\({b_2} = - 3{b_1}\).

Step-by-step solution

Writing the matrix in the augmented form

Write the given matrix in the augmented form \(\left( {\begin{array}{*{20}{c}}A&b\end{array}} \right)\).

\(\left( {\begin{array}{*{20}{c}}2&{ - 1}&{{b_1}}\\{ - 6}&3&{{b_2}}\end{array}} \right)\)

Step 2:Applying the row operation

Perform an elementary row operationto produce the first augmented matrix.

Write the sum of 3 times row one and row two in row two.

\(\left( {\begin{array}{*{20}{c}}2&{ - 1}&{{b_1}}\\0&0&{{b_2} + 3{b_1}}\end{array}} \right)\)

Converting the matrix into the equation form

To obtain the solution of the vector equation, convert the augmented matrix into vector equations.

Write the obtained matrix,\(\left( {\begin{array}{*{20}{c}}2&{ - 1}&{{b_1}}\\0&0&{{b_2} + 3{b_1}}\end{array}} \right)\),in the equation notation.

\(\begin{array}{c}2{x_1} - {x_2} = {b_1}\\0 = {b_2} + 3{b_1}\end{array}\)

Showing the equation \(Ax = b\) does not have a solution

The equation \(Ax = b\) is inconsistent when \(3{b_1} + {b_2}\) is non-zero.

Hence,the equation \(Ax = b\) does not have a solution.

Determining whether the equation \(Ax = b\) has a solution

The set \({\mathop{\rm b}\nolimits} \) for which the equation\(Ax = b\) is consistent can be a line passing through the origin for all the set of points \(\left( {{b_1},{b_2}} \right)\) that satisfy \({b_2} = - 3{b_1}\).

Thus, the equation \(Ax = b\) is consistent for \({b_2} = - 3{b_1}\).