Question

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

17. \(\left[ {\begin{array}{*{20}{c}}2&3&h\\4&6&7\end{array}} \right]\)

Short answer

The value of \(h\) is \(\frac{7}{2}\).

Step-by-step solution

Apply row operation

A basic principle states that row operations do not affect the solution set of alinear system. Perform an elementary row operation to produce the first augmented matrix.

Apply the sum of row 2 and \( - 2\)times row 1 at row 2.

\(\left[ {\begin{array}{*{20}{c}}2&3&h\\0&0&{7 - 2h}\end{array}} \right]\)

Convert the matrix into the equation

To determine the value of \(h\), you have to convert the augmented matrix into a system of equations.

Write the obtained matrix into the equation notation.

\(\begin{array}{c}2{x_1} + 3{x_2} = h\\{x_1}\left( 0 \right) + {x_2}\left( 0 \right) = 7 - 2h\end{array}\)

Obtain the value of h

A system of linear equations has aunique solution and isconsistent if the numbers of nonzero rows and the number of variables are equal.

Now, obtain the value of \(h\) by equating \(7 - 2h\) to 0.

\(\begin{aligned}{c}7 - 2h = 0\\2h = 7\\h = \frac{7}{2}\end{aligned}\)

Thus, the value of \(h\) is \(\frac{7}{2}\).