# Matrix equation Ax=b

### Matrix equation Ax=b

#### Lessons

If $A$ is an $m \times n$ matrix with columns $a_1$,…,$a_n$, and if $x$ is in $\Bbb{R}^n$, then the product of $A$ and $x$ is the linear combination of the columns in A using the corresponding entries in $x$ as weights. In other words,

If we were to say that $Ax=b$, then basically:
$a_1 x_1+\cdots+a_n x_n=b$

which we see b is a linear combination of $a_1,\cdots,a_n$. You will see questions where we have to solve for the entries of $x$ again, like last section.

We say that an equation in the form of $Ax=b$ is a matrix equation.

Properties of $Ax$
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. $A(cu)=c(Au)$
• 1.
Matrix Equation Ax=b Overview:
a)
Interpreting and Calculating $Ax$
• Product of $A$ and $x$
• Multiplying a matrix and a vector
• Relation to Linear combination

b)
Matrix Equation in the form $Ax=b$
• Matrix equation form

c)
Solving x
• Matrix equation to an augmented matrix
• Solving for the variables

d)
Properties of Ax
• Scalar property

• 2.
Computing Ax
Compute the following. If it cannot be computed, explain why:
a)

b)

c)

• 3.
Converting to Matrix Equation and Vector Equation
Write the given systems of equations as a vector equation, and then to a matrix equation.
$6x_1+2x_2-3x_3=1$
$2x_1-5x_2+x_3=4$
$-x_1-2x_2-7x_3=5$

• 4.
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.
a)

b)

• 5.
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.

• 6.
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:
$A[(2u-3v)(2u+3v)]=4A(u^2 )-9A(v^2)$