matrix multiplication

[Tiger-T]

Let A =[1 2]
[3 1], X = [X with a tiny lower 1]
[X with a tiny lower 2]
, and B = [-4]
[12].

Show that the equation AX = B represents a linear system of two equations in two unknowns. Solve the system and substitute into the matrix equation to check your results.

I have figured out how to multiply a matrix, however I have not been able to figure out what X1 and X2 equal. I have figured out that X means the matrix is a single column, but I just need to know the values of X1 and X2 to multiply and solve the problem.

 

[soroban]

Hello, Tiger-T!

\(\displaystyle \text{Let }\:A \:=\:\begin{bmatrix}1 & 2 \\3 & 1\end{bmatrix} \quad X \:=\:\begin{bmatrix}x_1 \\ x_2\end{bmatrix} \quad B \:=\:\begin{bmatrix}\text{-}4 \\ 12 \end{bmatrix}\)

\(\displaystyle \text{Show that the equation }AX \,=\, B\,\text{ represents a linear system of two equations in two unknowns.}\)

\(\displaystyle \text{Solve the system and substitute into the matrix equation to check your results.}\)

\(\displaystyle \text{We are given: }\:AX \:=\:B\)

\(\displaystyle \text{This means: }\:\begin{bmatrix}1 & 2 \\ 3 & 1\end{bmatrix}\,\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} \;=\; \begin{bmatrix}\text{-}4 \\ 12\end{bmatrix}\)

. . \(\displaystyle \text{which gives us: }\;\begin{Bmatrix} x_1 + 2x_2 &=& \text{-}4 \\ 3x_1 + x_2 &=& 12 \end{Bmatrix}\)


Are you saying that you can't solve the system?

 

[Tiger-T]

Soroban, I thank you for your help. I do not know if that solves the problem or not. I have no clue on the meaning of the X1 and the X2. Although, your solution looks impressive!!!

 

[mmm4444bot]

Soroban did only the first part, so, no, the information in his post does not "solve the problem".

I mean, Soroban followed the given instruction that reads, "Show that the equation AX = B represents a linear system of two equations in two unknowns."

Now you need to "Solve the system and substitute into the matrix equation to check your results."

Soroban asked you if you know how to do this. You seem to have ignored his question. Do you still need help?

Here are some comments on the "little numbers".

x[sub:30xes9f9]1[/sub:30xes9f9] and x[sub:30xes9f9]2[/sub:30xes9f9] are "subscripted" variables.

.

Those little numbers are the subscripts. This notation is simply a method of numbering different variables represented by the same letter of the alphabet.

.

You can think of x[sub:30xes9f9]1[/sub:30xes9f9] as "the first variable called x".

.

You can think of x[sub:30xes9f9]2[/sub:30xes9f9] as "the second variable called x".

.

These subscripted names show that x[sub:30xes9f9]1[/sub:30xes9f9] and x[sub:30xes9f9]2[/sub:30xes9f9] are two different variables.

.

Most of time, exercises use a different letter of the alphabet for each variable, like x, y, and z. Other times, we use the same letter for all of the variables and distinguish them with subscripts.

If it helps, you are free to rename the variables, solve the system using these new names, and then switch back to the given subscripted names.

For example, you could change the names as follows.

Let x[sub:30xes9f9]1[/sub:30xes9f9] be called x

.

Let x[sub:30xes9f9]2[/sub:30xes9f9] be called y

.

With these name changes, the system in this exercise becomes:

x + 2y = -4

3x + y = 12

Solve this system, and you'll have the values for x and y. Switch back to the given variable names.

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x[sub:30xes9f9]1[/sub:30xes9f9] = the solution you find for x

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x[sub:30xes9f9]2[/sub:30xes9f9] = the solution you find for y

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In time, you should learn to think symbolically, and use the given symbols. Again, they are nothing more than names. You could call the variables Jack and Jill, if you were so inclined. The important thing is understanding the distinction in your mind.