Calculating 4A + 3B A Step-by-Step Matrix Guide

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of matrix operations. Specifically, we're going to tackle a classic problem: finding the result of 4A + 3B given two matrices, A and B. Don't worry if you're new to this – we'll break it down step by step, making sure you grasp every concept along the way. So, grab your calculators (or just your brain!), and let's get started!

Understanding the Basics: Matrices A and B

Before we jump into the calculation, let's clearly define our players: the matrices A and B. In this case, we have:

A = \begin{bmatrix}
    1 & 6 \\
    7 & 2
\end{bmatrix}

B = \begin{bmatrix}
    -4 & 1 \\
    7 & -1
\end{bmatrix}

As you can see, both A and B are 2x2 matrices. This means they have two rows and two columns. The numbers inside the brackets are called elements or entries of the matrix. Understanding the dimensions and elements of a matrix is crucial for performing operations like scalar multiplication and addition, which we'll be using shortly. Think of matrices as organized tables of numbers, and just like tables, their structure dictates how we can manipulate them. For example, you can only add matrices that have the same dimensions. Trying to add a 2x2 matrix to a 3x2 matrix is like trying to fit a square peg in a round hole – it just won't work! So, remember, dimensions matter in the world of matrices. Now that we've got a good grasp of our matrices A and B, let's move on to the first operation: scalar multiplication.

Step 1: Scalar Multiplication - Finding 4A and 3B

The first step in solving 4A + 3B is to perform scalar multiplication. This simply means multiplying each element of a matrix by a constant value (the scalar). In our case, we need to find 4A and 3B. Let's start with 4A. To find 4A, we multiply each element of matrix A by 4:

4A = 4 * \begin{bmatrix}
    1 & 6 \\
    7 & 2
\end{bmatrix} = \begin{bmatrix}
    4*1 & 4*6 \\
    4*7 & 4*2
\end{bmatrix} = \begin{bmatrix}
    4 & 24 \\
    28 & 8
\end{bmatrix}

See? It's pretty straightforward! We just distributed the scalar 4 to every element inside the matrix. Now, let's do the same for 3B. We multiply each element of matrix B by 3:

3B = 3 * \begin{bmatrix}
    -4 & 1 \\
    7 & -1
\end{bmatrix} = \begin{bmatrix}
    3*(-4) & 3*1 \\
    3*7 & 3*(-1)
\end{bmatrix} = \begin{bmatrix}
    -12 & 3 \\
    21 & -3
\end{bmatrix}

Awesome! We've successfully found 4A and 3B. Scalar multiplication is a fundamental operation in linear algebra, and it's used extensively in various applications, from computer graphics to solving systems of equations. Think of it as scaling the entire matrix up or down by a certain factor. Just like multiplying a vector by a scalar changes its magnitude, multiplying a matrix by a scalar changes the magnitude of all its elements. With 4A and 3B in hand, we're now ready for the final step: adding these two matrices together.

Step 2: Matrix Addition - Combining 4A and 3B

Now comes the exciting part: matrix addition! To add two matrices, they must have the same dimensions (which, thankfully, 4A and 3B do – they're both 2x2). We simply add the corresponding elements of the matrices together. That is, we add the element in the first row and first column of 4A to the element in the first row and first column of 3B, and so on. Let's put it into action:

4A + 3B = \begin{bmatrix}
    4 & 24 \\
    28 & 8
\end{bmatrix} + \begin{bmatrix}
    -12 & 3 \\
    21 & -3
\end{bmatrix} = \begin{bmatrix}
    4 + (-12) & 24 + 3 \\
    28 + 21 & 8 + (-3)
\end{bmatrix}

Now, let's perform the additions:

= \begin{bmatrix}
    -8 & 27 \\
    49 & 5
\end{bmatrix}

And there you have it! We've successfully found 4A + 3B. Matrix addition is a powerful tool for combining information represented in matrix form. It's used in various fields, such as image processing (where images can be represented as matrices) and network analysis (where relationships between entities can be represented as matrices). Think of it as stacking the two matrices on top of each other and adding the numbers in each corresponding cell. The result is a new matrix that represents the combined information from the original matrices. So, with our final answer in hand, let's recap what we've learned.

Final Result

Therefore,

4A + 3B = \begin{bmatrix}
    -8 & 27 \\
    49 & 5
\end{bmatrix}

Conclusion: You've Mastered Matrix Operations!

Congratulations, guys! You've successfully navigated the world of matrix operations and found the solution to 4A + 3B. We started by understanding the basics of matrices, then moved on to scalar multiplication and matrix addition. These are fundamental concepts in linear algebra, and mastering them will open doors to more advanced topics and applications. Remember, practice makes perfect! The more you work with matrices, the more comfortable you'll become with these operations. So, keep exploring, keep learning, and keep pushing your mathematical boundaries. You've got this!

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. Given C = [[2, -1], [0, 3]] and D = [[-5, 4], [1, -2]], find 2C - D.
2. If E = [[1, 2], [3, 4]] and F = [[5, 6], [7, 8]], find 3E + 2F.
3. Let G = [[-2, 0], [1, 5]] and H = [[4, -3], [-1, 2]], find 5G - 4H.

Work through these problems, and you'll be a matrix operation pro in no time! Remember to follow the steps we outlined earlier: scalar multiplication first, then matrix addition or subtraction. Good luck, and happy calculating!

Further Exploration

If you're eager to delve deeper into the world of matrices, here are some topics you might find interesting:

• Matrix Multiplication: This is another fundamental operation that involves multiplying two matrices together. It's a bit more complex than addition, but it's incredibly powerful.
• Determinants and Inverses: These are important concepts that are used to solve systems of linear equations and analyze the properties of matrices.
• Eigenvalues and Eigenvectors: These are special values and vectors associated with a matrix that reveal important information about its behavior.

Linear algebra is a vast and fascinating field with applications in many areas of science, engineering, and computer science. So, keep exploring, and you'll discover the amazing power of matrices!