Linear algebra problem phrasing

Question

I do know how to solve a system of equations with an augmented matrix. But I'm not sure if that's what I have to do here. Can somebody help me? enter image description here

Answer 1

Consider the linear system $$ Ax=b $$ where $A=(a_{ij})$ is a square matrix and $b=(b_{i})$ is a vector of appropriate dimensions. More explicitly, we can write this as $$ \begin{pmatrix}a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \cdots & a_{nn} \end{pmatrix}\begin{pmatrix}x_{1}\\ \vdots\\ x_{n} \end{pmatrix}=\begin{pmatrix}b_{1}\\ \vdots\\ b_{n} \end{pmatrix} $$ or equivalently as the system of equations $$ \begin{cases} a_{11}x_{1}+\cdots+a_{1n}x_{n}=b_{1}\\ \vdots\\ a_{n1}x_{1}+\cdots+a_{nn}x_{n}=b_{n}. \end{cases} $$

Row reduction is a procedure in which various operations are performed on the linear system to arrive at an (equivalent) upper triangular linear system of the form $$ \begin{pmatrix}a_{11}^{\prime} & \cdots & a_{1n}^{\prime}\\ & \ddots & \vdots\\ & & a_{nn}^{\prime} \end{pmatrix}\begin{pmatrix}x_{1}\\ \vdots\\ x_{n} \end{pmatrix}=\begin{pmatrix}b_{1}^{\prime}\\ \vdots\\ b_{n}^{\prime} \end{pmatrix}. $$

Note that row reduction does not modify the vector $x$. Therefore, it is reasonable to represent row reduction in the following way: $$ \begin{pmatrix}a_{11} & \cdots & a_{1n} & b_{1}\\ \vdots & \ddots & \vdots & \vdots\\ a_{n1} & \cdots & a_{nn} & b_{n} \end{pmatrix}\xrightarrow{\text{row reduce}}\begin{pmatrix}a_{11}^{\prime} & \cdots & a_{1n}^{\prime} & b_{1}^{\prime}\\ & \ddots & \vdots & \vdots\\ & & a_{nn}^{\prime} & b_{n}^{\prime} \end{pmatrix}. $$ The matrices above are referred to as "augmented" in that they also carry information from the right-hand sides $b$ and $b^{\prime}$ of the linear systems above.

The question on your assignment is testing your knowledge of these concepts, and for example, asking you to go back and forth between the notations.