Question

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

18. \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\5&h&{ - 7}\end{array}} \right]\)

Short answer

The system is consistent if \(h \ne - 15\).

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 \( - 5\) times of row 1 at row 2.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\0&{15 + h}&3\end{array}} \right]\)

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.

\(\begin{array}{c}{x_1} - 3{x_2} = - 2\\{x_1}\left( 0 \right) + {x_2}\left( {15 + h} \right) = 3\end{array}\)

Obtain the value of h

A system of linear equations has a unique solution and is consistent if the numbers of nonzero rows and the number of variables are equal. The row-reduced echelon form of the matrix for the system of linear equations is \(\left[ {\begin{array}{*{20}{c}}1&0&2\\0&1&3\end{array}} \right]\)

The given system is consistent. Thus, this shows that \(15 + h \ne 0\) must be nonzero. Otherwise, the second row would correspond to an equation of the form 0 = b, where b would be nonzero. The system is consistent if \(h \ne - 15\).

Thus, the value of \(h\) is \(h \ne - 15\).