Question

Suppose \(AD = {I_m}\) (the \(m \times m\) identity matrix). Show that for any b in \({\mathbb{R}^m}\), the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) has a solution. (Hint: Think about the equation \(AD{\mathop{\rm b}\nolimits} = {\mathop{\rm b}\nolimits} \).) Explain why Acannot have more rows than columns.

Short answer

Amust have at least as many columns as rows because each pivot is in a different row.

Step-by-step solution

Show the equation \(Ax = b\) has a solution

Choose any b in \({\mathbb{R}^m}\). \(ADb = {I_m}b = b\), according to the hypothesis. Rewrite the equation as \(A\left( {D{\mathop{\rm b}\nolimits} } \right) = {\mathop{\rm b}\nolimits} \). Therefore, the vector \(x = D{\mathop{\rm b}\nolimits} \) satisfies the equation \(Ax = {\mathop{\rm b}\nolimits} \). This establishes that the equation \(Ax = {\mathop{\rm b}\nolimits} \) has a solution for each b in \({\mathbb{R}^m}\).

Explanation of A cannot have more rows than columns

Theorem 4states that \(A\) be a \(m \times n\) matrix. Then, \(A\) has a pivot position in every row.

Ahas a pivot position in every row according to theorem 4.Thus, Amust have at least as many columns as rows because each pivot is in a different row.