Weekly Math Problem! #45

Diagonalization. I want to go over some linear algebra this week so, here is a problem to help me with that. I landed on diagonalization of a matrix because, for some reason, it stuck with me. I can't really explain why. Check out my other WMPs that involved working with matrices--WMP! #4 and WMP! #14. I don't have much else to say (due to tiiirrreeeddd-ness) 😩. 

⚠️Many steps involved.⚠️To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to perform row operations
        ✔️     How to find the determinant of a matrix
        ✔️     How to solve a quadratic equation
        ✔️     How to multiply matrices
        ✔️     How to find the inverse of a matrix

All calculations can be done without a calculator but, feel free to use a matrix calculator to check your work. Ready?? 😁🥴 WMP! #45 says...

Happy solving!

Check back on Friday, April 9th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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I'm just gonna go 'head and get into this calculation...computation...situation...

😅

Step 1. Find the Characteristic Polynomial

Step 2. Find the Eigenvalues 

Step 3. Find the Eigenspaces and Eigenvectors 

Step 4. Define Invertible Matrix, F, and Diagonal Matrix, D 

Step 5. Complete the Diagonalization

 

(Check the Diagonalization)

Step 6. Complete the Diagonalization

(Final Check)

**This plot was created using Desmos' Matrix Calculator.

WHEW!!! That was a lot. (You can't say I didn't warn you. With that...I think I'm gonna take it easy next week. 🥴)

▪️ Did you survive this calculation?

▪️ Have you ever taken Linear Algebra?

▪️ Leave your responses down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #46.

Cheers!

The Younge Lady

Weekly Math Problem! #34

Inverse of a Function. I have different ways of choosing problems to solve on this blog. Sometimes I go through books and notes I have. Other times, I use Google. 😁 This week, Google won as my method due to a lack of time. 🤷🏿‍♀️ What can I say? So, I decided to check out some precal topics and found a lighter problem to use. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the inverse of a function

That's it? Yup...that's it. WMP! #34 wants us to...

Happy solving!

Check back on Friday, January 22nd for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Let's go right on ahead and jump in this thing! 

Here comes the fun 🤪 part! (At least I think it's fun. 🤷🏿 ) The equation right above here is a literal equation. We are going to solve it but, the answer we get will not be a number. By placing x over 1, it can be more clearly seen that the above equation is a proportion.

We're just about done. Just one more thing...a mere formality:

That's it! We've found the inverse function, f -1. 🤸🏿‍♀️ Yay! Like the original function, the inverse is also a rational function, with a variable present in the denominator. That means we cannot use, or plug-in, any value for x that we like. To continue with the second part of the question, we will find the domain for f -1 by identifying which value of x will cause the function to be undefined. I did something slightly unconventional to find the value. Instead of setting the denominator equal to zero, I set the denominator not equal zero and proceeded normally:

Now, I can write the domain formally. I don't really have a preference for notation, so I did it two ways:

 

You know I wouldn't end this blog post without graphing the functions, right? Okay, great! (I tried something a little different with the color scheme. What do you think?) When a function and its inverse are plotted on the same set of axes, they will be mirror images of each other. The mirror is the line y = x.

**This plot was created using Geogebra's Graphing Gaclulator.

▪️ Let me know what you thought about this week's problem down below.

Thanks for solving with me this week!

Up next...WMP! #35

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Cheers!

The Younge Lady