Matrix Multiplication: Can These Matrices Be Multiplied?

Are you ready to dive into the world of matrices and matrix multiplication? This guide will help you determine if two matrices can be multiplied. We'll explore the rules and provide a clear "Yes" or "No" answer based on the dimensions of the matrices provided. Let's get started and unravel the magic behind matrix multiplication! In this article, we'll focus on whether specific matrices can be multiplied, providing a straightforward answer to the question: Can the following matrices be multiplied?

$\left[\begin{array}{ccc}0 & 2 & 3 \\ 4 & 1 & 5\end{array}\right] \times\left[\begin{array}{cccc}0 & 2 & 1 & 3 \\ 5 & 7 & 4 & 1 \\ 6 & 2 & -2 & 0\end{array}\right]$

Understanding Matrix Multiplication

Before we can determine if the matrices can be multiplied, we need to understand the fundamental rule governing matrix multiplication. Matrix multiplication isn't always possible; it depends on the dimensions of the matrices involved. Think of matrix dimensions as the blueprint that tells us how many rows and columns a matrix has. The dimensions are represented as (rows x columns). For two matrices, A and B, to be multiplied (A x B), the number of columns in matrix A must be equal to the number of rows in matrix B. If this condition is met, we can proceed with the multiplication. If it's not met, the multiplication is undefined, and we cannot perform it.

The Dimensions Rule

The core principle to remember is the dimensions compatibility. Let's break it down:

• Matrix A: If matrix A has dimensions (m x n), where 'm' is the number of rows and 'n' is the number of columns.
• Matrix B: If matrix B has dimensions (p x q), where 'p' is the number of rows and 'q' is the number of columns.

For the multiplication A x B to be valid, the number of columns in A (n) must equal the number of rows in B (p). If n = p, then the resulting matrix will have dimensions (m x q).

Practical Example

Let's say we have:

• Matrix A: (2 x 3)
• Matrix B: (3 x 4)

Since the number of columns in A (3) is equal to the number of rows in B (3), we can multiply these matrices. The resulting matrix will have dimensions (2 x 4). If matrix A was (2 x 3) and matrix B was (2 x 4), we cannot multiply these matrices because the inner dimensions (3 and 2) don't match.

Applying the Rules: Checking the Given Matrices

Now, let's apply this knowledge to the matrices in question. We have two matrices, and we need to check if we can multiply them. Here's the breakdown:

• Matrix 1: $\left[\begin{array}{ccc}0 & 2 & 3 \\ 4 & 1 & 5\end{array}\right]$
• Matrix 2: $\left[\begin{array}{cccc}0 & 2 & 1 & 3 \\ 5 & 7 & 4 & 1 \\ 6 & 2 & -2 & 0\end{array}\right]$

Step-by-Step Analysis

1. Identify the Dimensions:

   • Matrix 1 has 2 rows and 3 columns, so its dimensions are (2 x 3).
   • Matrix 2 has 3 rows and 4 columns, so its dimensions are (3 x 4).

2. Check for Compatibility:

   • The number of columns in Matrix 1 is 3.
   • The number of rows in Matrix 2 is 3.
   • Since the number of columns in Matrix 1 equals the number of rows in Matrix 2 (3 = 3), the matrices can be multiplied.

3. Determine the Resulting Matrix Dimensions:

   • Because the inner dimensions match, the resulting matrix will have the dimensions of the outer dimensions: (2 x 4).

The Answer

Given the analysis above, we can confidently say that the matrices can be multiplied. The resulting matrix will have dimensions (2 x 4).

Step-by-Step Matrix Multiplication (Example)

To solidify your understanding, let's briefly walk through the process of matrix multiplication. This example will show you how to multiply two matrices. Suppose we have matrix A (2x2) and matrix B (2x2).

• Matrix A: $\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]$
• Matrix B: $\left[\begin{array}{cc}e & f \\ g & h\end{array}\right]$

The resulting matrix C (2x2) will be calculated as follows:

• C11 = (a * e) + (b * g)
• C12 = (a * f) + (b * h)
• C21 = (c * e) + (d * g)
• C22 = (c * f) + (d * h)

So, matrix C would look like:

$\left[\begin{array}{cc}(a * e) + (b * g) & (a * f) + (b * h) \\ (c * e) + (d * g) & (c * f) + (d * h)\end{array}\right]$

Applying to a Numerical Example

Let's use specific numbers:

• Matrix A: $\left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]$
• Matrix B: $\left[\begin{array}{cc}5 & 6 \\ 7 & 8\end{array}\right]$

Then, the matrix C will be calculated as follows:

• C11 = (1 * 5) + (2 * 7) = 5 + 14 = 19
• C12 = (1 * 6) + (2 * 8) = 6 + 16 = 22
• C21 = (3 * 5) + (4 * 7) = 15 + 28 = 43
• C22 = (3 * 6) + (4 * 8) = 18 + 32 = 50

Therefore, matrix C will be:

$\left[\begin{array}{cc}19 & 22 \\ 43 & 50\end{array}\right]$

Key Takeaways and Conclusion

In conclusion, understanding matrix multiplication is crucial in linear algebra. The ability to determine if matrices can be multiplied is the first step in solving many mathematical problems. Remember the key rule: The number of columns in the first matrix must equal the number of rows in the second matrix. If these dimensions align, the multiplication is possible. The resulting matrix's dimensions are formed by the outer dimensions of the original matrices.

Now you're equipped to determine if matrices can be multiplied. Keep practicing, and you'll become proficient in this essential skill. Matrix multiplication is fundamental in many areas of mathematics and computer science, so mastering it opens doors to more complex and exciting concepts. Matrix multiplication is not just about multiplying numbers; it's about representing transformations, solving systems of equations, and understanding the relationships between different datasets. This is a very important concept. So, go forth and explore the world of matrices, armed with the knowledge of how to multiply them! Remember that practice is key, and the more you work with matrices, the more comfortable and confident you'll become. Keep exploring and asking questions; that is how you learn. Keep up the good work!

Answer: Yes

For further exploration, you may find these resources helpful:

• Khan Academy's Linear Algebra: Khan Academy