Matrix Multiplication: Scalar Product and Properties of Matrices
Hello everyone and welcome to the “Futuros Sobresalientes” channel! I’m Miguel Fernández Collado, and today we’re diving into mathematics with a focus on the fascinating topic of matrix multiplication. If you aren’t subscribed yet, please do so, as it is free! You can also join the channel or support me on Patreon. If you find yourself needing private classes, be sure to check the link to our website in the description below.
The Basics: Scalar Product of Matrices
Today’s lesson is quite straightforward. We are going to explore the product of a matrix by a scalar. Let’s begin with two matrices:
- Matrix A: [3, -3, 0, 1, 2]
- Matrix B: [4, 2, 5, 1, 3]
We will perform the following multiplication: -2 * A. Here, the number -2 multiplies every element in matrix A:
| Matrix Element | Multiplication | Result |
|---|---|---|
| 3 | -2 * 3 | -6 |
| -3 | -2 * -3 | 6 |
| 0 | -2 * 0 | 0 |
| 1 | -2 * 1 | -2 |
| 2 | -2 * 2 | -4 |
After performing these multiplications, we find the resulting matrix:
- Resulting Matrix: [-6, 6, 0, -2, -4]
Another Example: Fraction Multiplication
Next, let’s consider 1/7 * B. This means we will multiply every element in matrix B by 1/7:
| Matrix Element | Multiplication | Result |
|---|---|---|
| 49 | 1/7 * 49 | 7 |
| -56 | 1/7 * -56 | -8 |
| 21 | 1/7 * 21 | 3 |
| 0 | 1/7 * 0 | 0 |
Thus, the resulting matrix after the multiplication is:
- Resulting Matrix: [7, -8, 3, 0]
Properties of Matrix Multiplication
Next, let’s explore some important properties of matrix operations. Understanding these properties is essential in both solving problems and simplifying calculations.
Distributive Property
1. If we have a number multiplying a sum of matrices:
k * (A + B) = kA + kB
2. If two numbers are added together and then multiplied by a matrix:
(a + b) * C = a * C + b * C
Let’s demonstrate this with an example using:
- Number A = 2
- Number B = -1
We can confirm the first property:
| Expression | Calculation | Result |
| 2 * (A + B) | 2 * [4 + 2, 3 + 5, 1 + 1, 0 + 3] | 2 * [6, 8, 2, 3] = [12, 16, 4, 6] |
Verifying the Second Property
Now, let’s check the validity of the second property:
| Expression | Calculation | Result |
| 2 * A + (-1) * A | 2 * [4, 3, 1, 0] + (-1) * [4, 3, 1, 0] | Resulting in a sum of elements confirming the property |
Practice Exercise for You!
Now, I challenge you with a practice exercise. Calculate the result of:
- 2 * A – 3 * B
Feel free to share your answers in the comments below!
Thank you for joining today’s mathematics lesson! I hope you found it as enjoyable as I did. Remember, if you are looking for online math help or need private classes, don’t hesitate to explore our website linked in the description. Join our channel for more math tutorials and similar content.
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