Weekly Math Problem #70

Eigenvalues. This week, linear algebra came to mind. The last time I did a linear algebra related problem on this blog was in WMP #45. Check it out...it may help you with this week's problem. 👀 *clears throat*. Anyhow, eigenvalues are cool and have lots of applications. For more info, check out the Applications of Eigenvalues and Eigenvectors page from Interactive Mathematics website. 

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

        ✔️     How to find the determinant of a 2x2 matrix
        ✔️     How to solve a quadratic equation
             

WMP! #70 wants us to...

Happy solving!

Check back on Saturday, 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 get it started...🎶

The process to finding the eigenvalues of a 2x2 matrix requires working with a polynomial called the characteristic polynomial. Ours will be a quadratic polynomial generated from finding the determinant of (Y - λI2), where I2 is the 2x2 identity matrix.

Now that we have the characteristic polynomial, we need to find it roots. The roots of the characteristic polynomial are the eigenvalues of the matrix. Set the characteristic polynomial equal to zero and solve. Ours is factorable, but if it wasn't, then you'd have the option to use other methods for solving a quadratic equation.

The eigenvalues have been found!

▪️ Hopefully you haven't found the process to finding the eigenvalues of a 2x2 matrix daunting.

▪️ Let me know what you thought about this week's problem in the comments section. 

Thank you for solving with me this week. 😊
📗 WMP! #71 is up next. 

Cheers!

The Younge Lady

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