Question

Let A be a \(3 \times 2\) matrix. Explain why the equation \(A{\bf{x}} = {\bf{b}}\) cannot be consistent for all b in \({\mathbb{R}^3}\). Generalize your argument to the case of an arbitrary A with more rows than columns.

Short answer

The equation \(A{\bf{x}} = {\bf{b}}\) is not consistent because one of the rows does not have pivot positions.

Step-by-step solution

Writing the definition of \(A{\bf{x}}\)

The column of matrix \(A\) is represented as \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\), and vector x is represented as \(\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\).

According to the definition, the weights in a linear combination of matrix A columns are represented by the entries in vector x.

The matrix equation as a vector equation can be written as shown below:

\(\begin{array}{c}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right)\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\b = {x_1}{a_1} + {x_2}{a_2} + \cdots + {x_n}{a_n}\end{array}\)

The number of columns in matrix \(A\) should be equal to the number of entries in vector x so that \(A{\bf{x}}\) can be defined.

Writing the condition for a consistent solution

Consider an \(m \times n\) ordered matrix A. Let \(m = n\), which means the number of rows is equal to the number of columns.

For \(m = n\), the matrix has maximum \(n\) pivot positions that can be filled by \(m\) rows. The equation \(A{\bf{x}} = {\bf{b}}\) is consistent in this case.

Again, consider an \(m \times n\) ordered matrix A. Let \(m > n\), which means the number of rows is greater than the number of columns.

For \(m > n\), the matrix has maximum \(n\) pivot positions that cannot be filled by \(m\) rows. So, the equation \(A{\bf{x}} = {\bf{b}}\) is not consistent.

Checking the consistency for \(3 \times 2\) matrix

In a \(3 \times 2\) matrix, the number of rows is greater than the number of columns, i.e., \(m > n\). So, the matrix has a maximum of two pivot columns and two pivot positions.

Checking the consistency for \(3 \times 2\) matrix

In a \(3 \times 2\) matrix, two pivot positions are not enough to cover three rows

(as \(3 > 2\)). So, one of the rows does not have a pivot position.

It means the matrix cannot be consistent for all bin \({\mathbb{R}^3}\).

Thus, the equation \(A{\bf{x}} = {\bf{b}}\) is not consistent.