Suppose Ais a \(3 \times n\) matrix whose columns span \({\mathbb{R}^3}\). Explain how to construct an \(n \times 3\) matrix Dsuch that \(AD = {I_3}\).

\(A\)is a \(m \times n\) matrix. If Bis a \(n \times p\) matrix with columns \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _p}\), the product ABis the \(m \times p\) matrix whose columns are \(A{{\mathop{\rm b}\nolimits} _1},...,A{{\mathop{\rm b}\nolimits} _p}\). That is, \(AB = A\left( {\begin{aligned}{*{20}{c}}{{b_1}}&{{b_2}}&{{b_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{A{b_1}}&{A{b_2}}&{A{b_3}}\end{aligned}} \right)\) .

Consider \({I_3} = \left( {\begin{aligned}{*{20}{c}}{{e_1}}&{{e_2}}&{{e_3}}\end{aligned}} \right)\) and \(D = \left( {\begin{aligned}{*{20}{c}}{{d_1}}&{{d_2}}&{{d_3}}\end{aligned}} \right)\). According to the definition of AD, the equation \(AD = {I_3}\) is equivalent to the three equations \(A{d_1} = {e_1}\), \(A{d_2} = {e_2}\), and \(A{d_3} = {e_3}\).

Since the columns of Aspan \({\mathbb{R}^3}\), each of these equations has at least one solution. Choose one solution of each equation and use it for the columns of D. Thus, \(AD = {I_3}\).