Matrix Equations

Introduction

The Matrix Equation $Ax=b$.

Spans and Consistency

Let $A$ be an $m\times n$ matrix. The product $Ax=b$ is a matrix equation. Which statement is true?

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $x = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. Which of the following is $Ax$?

In one word: an equation of the form $Ax=b$, where $A$ is a matrix and $x$ is a vector of unknowns, is called a ______ equation.

Solve the matrix equation

Enter $x_1,x_2$ (for example: 1,2).

Solving $Ax = \begin{pmatrix}1\\0\end{pmatrix}$ gives $x_1 = \begin{pmatrix}1\\-1\end{pmatrix}$, and solving $Ax = \begin{pmatrix}0\\1\end{pmatrix}$ gives $x_2 = \begin{pmatrix}-2\\3\end{pmatrix}$. Find the first row of the $2\times2$ matrix $A$ and enter its two entries (for example: 1,2).

In your own words, state the single most important definition or theorem of this section, and explain what it says.