In Exercise 19 and 20, choose \({\rm{h}}\) and \({\rm{k}}\) such that the system has

a. no solution

b. unique solution

c. many solutions.

Give separate answers for each part.

20. \(\begin{array}{l}{x_1} + 3{x_2} = 2\\3{x_1} + h{x_2} = k\end{array}\)

Anaugmented matrixfor a system of equations is a matrix of numbers in which each row represents theconstants from one equation, and eachcolumn represents all thecoefficients for a single variable.

The augmented matrix for the given system of equations \({x_1} + 3{x_2} = 2\) and \(3{x_1} + h{x_2} = k\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}{\rm{1}}&3&{\rm{2}}\\3&h&k\end{array}} \right]\)

Perform an elementary row operationto produce the first augmented matrix.

Replace row 2 by adding -3 times row 1 to row 2.

\(\left[ {\begin{array}{*{20}{c}}{\rm{1}}&h&{\rm{2}}\\{\rm{0}}&{h - 9}&{k - 6}\end{array}} \right]\)

A system of linear equations has no solution if the system isinconsistent. The matrix’s row-reduced echelon form for an inconsistent system has a row with a nonzero number in the last column and zeros in all other columns.

(a) No solution:

There is no solution when \(h - 9 = 0\) and \(k - 6 \ne 0\). The matrix has a row with the nonzero number in the last column; \(h = 2\) and \(k \ne {\rm{8}}\).

The system isconsistent and has aunique solution if pivots are in columns 1 and 2 but not in the last column.

(b) Unique solution:

There is a unique solution when pivots are in columns 1 and 2 but not in column 3; \(h - 9 \ne 0\),i.e., \(h \ne 9\).

A system of linear equations has infinitely many solutions when it is consistent,and the number of variables is more than the number of nonzero rows in the row-reduced echelon form of the matrix.

(c) Many solutions:

There are many solutions when the number of variables is more than the number of nonzero rows in the row-reduced matrix; when \(h - 9 = k - 6 = 0\), or when \({\rm{h = 9}}\) and \(k = 6\).