In Exercises 19–22, determine the value(s) of \['h'\]such that the matrix is the augmented matrix of a consistent linear system.

21. \[\left[ {\begin{array}{*{20}{c}}1&3&{ - 2}\\{ - 4}&h&8\end{array}} \right]\]

The given augmented matrix of a consistent linear system is as follows:

\[\left[ {\begin{array}{*{20}{c}}1&3&{ - 2}\\{ - 4}&h&8\end{array}} \right]\]

A basic principle states that row operations do not affect the solution set of a linear system.

To eliminate the first term of the second row,perform an elementary row operationon the augmented matrix \[\left[ {\begin{array}{*{20}{c}}1&3&{ - 2}\\{ - 4}&h&8\end{array}} \right]\],as shown below.

Add four times of the first row to the second row;i.e., \({R_2} \to {R_2} + 4{R_1}\).

\[\left[ {\begin{array}{*{20}{c}}1&3&{ - 2}\\{ - 4 + \left( {4 \times 1} \right)}&{h + \left( {4 \times 3} \right)}&{8 + \left( {4 \times - 2} \right)}\end{array}} \right]\]

\[\left[ {\begin{array}{*{20}{c}}1&3&{ - 2}\\0&{h + 12}&0\end{array}} \right]\]

For the system to beconsistent,it should have uniqueor infinitely many solutions.

Based on the above-obtained augmented matrix, the second equation is \[c{x_2} = 0\] in the equation notation form.

This equation has a solution for every value of c,which means thatthe system is consistent for all values of h.