Matrix Multiplication Result: Calculation And Explanation

by ADMIN 58 views

Hey guys! Let's dive into the fascinating world of matrix multiplication. In this article, we'll break down how to calculate the result of multiplying two matrices. We'll use a specific example to illustrate the process, making it super easy to understand. Whether you're a student tackling linear algebra or just curious about matrix operations, this guide is for you!

Understanding Matrix Multiplication

Before we jump into the calculation, let's quickly recap what matrix multiplication is all about. Multiplying matrices isn't as straightforward as multiplying regular numbers. The dimensions of the matrices matter, and the process involves a series of dot products between rows and columns. Specifically, to multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. If we have a matrix A with dimensions m x n and a matrix B with dimensions n x p, then the resulting matrix C will have dimensions m x p. Each element in matrix C is calculated by taking the dot product of the corresponding row in A and the corresponding column in B. This might sound a bit complex, but it becomes clear with an example, so let's get started!

The Problem: Multiplying Two Matrices

Okay, so the question we're tackling today is: What is the result of the matrix multiplication

[210−4][−352−1]\begin{bmatrix}2 & 1\\ 0 & -4\end{bmatrix} \begin{bmatrix}-3 & 5\\ 2 & -1\end{bmatrix}?

This problem involves multiplying two 2x2 matrices. That means the resulting matrix will also be a 2x2 matrix. We'll walk through each step of the calculation to make sure you understand exactly how to get the answer. Remember, it's all about taking the dot products of rows and columns!

Step-by-Step Calculation

Let's break down the matrix multiplication step by step. We'll label the matrices for clarity:

Matrix A: [210−4]\begin{bmatrix}2 & 1\\ 0 & -4\end{bmatrix}

Matrix B: [−352−1]\begin{bmatrix}-3 & 5\\ 2 & -1\end{bmatrix}

Our goal is to find Matrix C, where C = A * B.

Calculating the First Element (C₁₁)

The first element of the resulting matrix C, denoted as C₁₁, is found by taking the dot product of the first row of Matrix A and the first column of Matrix B.

First row of A: [2, 1]

First column of B: [-3, 2]

The dot product is calculated as follows:

(2 * -3) + (1 * 2) = -6 + 2 = -4

So, C₁₁ = -4

Calculating the Second Element (C₁₂)

Next, we find the element C₁₂ by taking the dot product of the first row of Matrix A and the second column of Matrix B.

First row of A: [2, 1]

Second column of B: [5, -1]

The dot product is:

(2 * 5) + (1 * -1) = 10 - 1 = 9

Therefore, C₁₂ = 9

Calculating the Third Element (C₂₁)

Now, let's find C₂₁, which is the dot product of the second row of Matrix A and the first column of Matrix B.

Second row of A: [0, -4]

First column of B: [-3, 2]

The dot product is:

(0 * -3) + (-4 * 2) = 0 - 8 = -8

Hence, C₂₁ = -8

Calculating the Fourth Element (C₂₂)

Finally, we calculate C₂₂ by taking the dot product of the second row of Matrix A and the second column of Matrix B.

Second row of A: [0, -4]

Second column of B: [5, -1]

The dot product is:

(0 * 5) + (-4 * -1) = 0 + 4 = 4

Thus, C₂₂ = 4

The Resulting Matrix

After performing all the calculations, we can now construct the resulting matrix C:

Matrix C = [−49−84]\begin{bmatrix}-4 & 9\\ -8 & 4\end{bmatrix}

So, the result of the matrix multiplication is:

[210−4][−352−1]\begin{bmatrix}2 & 1\\ 0 & -4\end{bmatrix} \begin{bmatrix}-3 & 5\\ 2 & -1\end{bmatrix} = [−49−84]\begin{bmatrix}-4 & 9\\ -8 & 4\end{bmatrix}

Key Concepts in Matrix Multiplication

Matrix multiplication might seem intimidating at first, but it’s all about following the steps carefully. Remember these key concepts:

  • Dimensions Matter: For matrix multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix.
  • Dot Product: Each element in the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.
  • Order Matters: Matrix multiplication is not commutative, meaning A * B is generally not the same as B * A.
  • Step-by-Step: Breaking down the calculation into smaller steps (like we did above) makes the process much easier to manage.

Understanding these concepts will help you tackle a wide range of matrix multiplication problems. Keep practicing, and you'll become a pro in no time!

Common Mistakes to Avoid

When you're learning matrix multiplication, it’s easy to make a few common mistakes. Here are some pitfalls to watch out for:

  • Incorrect Dimensions: Always double-check that the dimensions of the matrices are compatible for multiplication. If the number of columns in the first matrix doesn't equal the number of rows in the second, you can't multiply them.
  • Mixing Rows and Columns: Make sure you're taking the dot product of the correct row and column. It’s easy to get mixed up, especially when you're working with larger matrices. Labeling the rows and columns can help.
  • Arithmetic Errors: Simple arithmetic mistakes in the dot product calculations can throw off your entire result. Take your time and double-check your math.
  • Forgetting the Order: Remember that matrix multiplication isn't commutative. Don't assume that A * B is the same as B * A. Always stick to the correct order.

By being aware of these common mistakes, you can avoid them and ensure your matrix multiplication calculations are accurate.

Real-World Applications of Matrix Multiplication

You might be wondering,