Answer to Question #102058 in Linear Algebra for Alam Sha

Question

Suppose that A is a 3x3 matrix and that b is a 3x1 matrix. Assuming that
the matrix equation Ax = b has infinitely many solutions, write down all the possible
reduced row-echelon forms of the augmented matrix (A | b). You may use ‘*’ to
represent an unknown number.

Answer 1

Consider the matrix

$A=\begin{pmatrix} a_{11} &a_{12}&a_{13} \\ a_{21} &a_{22}&a_{23} \\ a_{31} &a_{32}&a_{33} \\ \end{pmatrix}$

$X=\begin{pmatrix} x_1 \\ x_2\\ x_3 \end{pmatrix}$

$b=\begin{pmatrix} b_1 \\ b_2\\ b_3 \end{pmatrix}$

Subject to the problem the matrix equation

$Ax = b$

has infinitely many solutions then $det A=0, rank(A)<3$.

The matrix equation $Ax = b$ :

$\left\{\begin{matrix} a_{11}x_1+ a_{12}x_2+a_{13}x_3=b_1\\ a_{21}x_1+ a_{22}x_2+a_{23}x_3=b_2\\ a_{31}x_1+ a_{32}x_2+a_{33}x_3=b_3\\ \end{matrix}\right.$

Consider the matrix

$(A|b)=\begin{pmatrix} a_{11} &a_{12}&a_{13} &|b_1\\ a_{21} &a_{22}&a_{23} &|b_2\\ a_{31} &a_{32}&a_{33} &|b_3\\ \end{pmatrix}$

Suppose $a_{11}\neq 0, rank(A)=2$

$(A|b) \leftrightarrow\begin{pmatrix} 1 &\frac{a_{12}}{a_{11}}&\frac{a_{13}}{a_{11}} &|\frac{b_1}{a_{11}}\\ a_{21} &a_{22}&a_{23} &|b_2\\ a_{31} &a_{32}&a_{33} &|b_3\\ \end{pmatrix} \leftrightarrow\\ \begin{matrix} R_2+R_1(-a_{21}) \\ R_3+R_1(-a_{31}) \end{matrix}\\ \leftrightarrow \begin{pmatrix} 1 &*&* &|*\\ 0 &c_{22}&c_{23} &|c_2\\ 0 &c_{32}&c_{33} &|c_3\\ \end{pmatrix}$

$c_{22}\neq 0, rg(A)=2$

perform operations over rows

$\frac{R_2}{c_{22}}\\ R_3+R_2(-c_{32})$

$(A|b)\leftrightarrow \begin{pmatrix} 1 &*&* &|*\\ 0 &1&* &|*\\ 0 &0&0 &|0\\ \end{pmatrix}$

Similarly, if $rank(A)=1, a_{11}\neq 0$, then

$(A|b)\leftrightarrow \begin{pmatrix} 1 &*&* &|*\\ 0 &0&0 &|0\\ 0 &0&0 &|0\\ \end{pmatrix}$