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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 \({x_1} + h{x_2} = 2\) and \(4{x_1} + 8{x_2} = k\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}{\rm{1}}&h&{\rm{2}}\\{\rm{4}}&{\rm{8}}&k\end{array}} \right]\)

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.

Replace row 2 by adding -4 times row 1 to row 2.

\(\left[ {\begin{array}{*{20}{c}}{\rm{1}}&h&{\rm{2}}\\{\rm{0}}&{8 - 4h}&{k - 8}\end{array}} \right]\)

A system of linear equations has no solution if the system is inconsistent. The matrix鈥檚 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:

When \(8 - 4h = 0\) and \(k - 8 \ne 0\), the matrix has a row with the nonzero number in the last column. There is no solution when \(h = 2\) and \(k \ne {\rm{8}}\).

The system is consistent and has a unique 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; when \({\rm{8 - 4}}h \ne 0\),i.e., \(h \ne {\rm{2}}\).

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 \({\rm{8 - 4}}h = k{\rm{ - 8 = 0}}\), or when \(h{\rm{ = 2}}\) and \(k{\rm{ = 8}}\).