# Matrix equation Ax=b

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##### Intros
###### Lessons
1. Matrix Equation Ax=b Overview:
2. Interpreting and Calculating $Ax$
• Product of $A$ and $x$
• Multiplying a matrix and a vector
• Relation to Linear combination
3. Matrix Equation in the form $Ax=b$
• Matrix equation form
4. Solving x
• Matrix equation to an augmented matrix
• Solving for the variables
5. Properties of Ax
• Scalar property
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##### Examples
###### Lessons
1. Computing Ax
Compute the following. If it cannot be computed, explain why:
2. Converting to Matrix Equation and Vector Equation
Write the given systems of equations as a vector equation, and then to a matrix equation.
$6x_1+2x_2-3x_3=1$
$2x_1-5x_2+x_3=4$
$-x_1-2x_2-7x_3=5$
1. Solving the Equation $AX=b$
Write the augmented matrix for the linear system that corresponds to the matrix equation $Ax=b$. Then solve the system and write the solution as a vector.
2. Ax=b with unknown b terms
Let and . Show that the matrix equation $Ax=b$ does have solutions for some $b$, and no solution for some other $b$'s.
1. Understanding Properties of Ax
Recall that the properties of the matrix-vector product Ax is:

If $A$ is an $m \times n$ matrix, $u$ and $v$ are vectors in $\Bbb{R}^n$, $c$ is a scalar, then:
1. $A(u+v)=Au+Av$
2. $A(cu)=c(Au)$

Using these properties, show that:
$A[(2u-3v)(2u+3v)]=4A(u^2 )-9A(v^2)$