Matrix Multiplication: Find (AB)₂₃ Simply

by Luna Greco 42 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of matrix multiplication. Specifically, we're going to break down how to find a particular entry in the product of two matrices. Our mission? To determine the entry in the second row and third column, denoted as (AB)₂₃, of the product of matrices A and B, where A and B are given as:

A=[10−235−1],B=[2−10104−531]A=\begin{bmatrix}1 & 0 & -2 \\ 3 & 5 & -1\end{bmatrix}, B=\begin{bmatrix}2 & -1 & 0 \\ 1 & 0 & 4 \\ -5 & 3 & 1\end{bmatrix}

So, grab your calculators (or your mental math muscles) and let's get started!

Understanding Matrix Multiplication

Before we jump into the calculation, let's quickly recap the basics of matrix multiplication. Remember, multiplying matrices isn't as straightforward as multiplying numbers. It involves a specific row-by-column process.

The Row-by-Column Rule

The key thing to remember about matrix multiplication is that you multiply the elements of the rows of the first matrix by the elements of the columns of the second matrix. To get the entry in the i-th row and j-th column of the product matrix, you take the dot product of the i-th row of the first matrix and the j-th column of the second matrix.

  • What does that mean exactly? It means you pair up the first element of the row with the first element of the column, the second element of the row with the second element of the column, and so on. Then, you multiply each of these pairs and add the results together. This sum becomes the entry in your resulting matrix.

Dimensions Matter

Also, a crucial point to remember is that you can only multiply matrices if their dimensions are compatible. If matrix A is an m x n matrix (m rows and n columns) and matrix B is a p x q matrix (p rows and q columns), then you can only multiply A and B if n = p. The resulting matrix will have dimensions m x q.

  • In our case, A is a 2 x 3 matrix and B is a 3 x 3 matrix. Since the number of columns in A (3) matches the number of rows in B (3), we can multiply them. The resulting matrix AB will be a 2 x 3 matrix.

Now that we've refreshed our understanding of matrix multiplication, we're ready to tackle the problem at hand.

Calculating (AB)₂₃ Step-by-Step

Okay, let's get down to business! We want to find the entry in the second row and third column of the product matrix AB, which we denote as (AB)₂₃. Following the row-by-column rule, this means we need to take the dot product of the second row of matrix A and the third column of matrix B.

Identifying the Correct Row and Column

First, let's identify the second row of A and the third column of B:

  • Second row of A: [3 5 -1]
  • Third column of B: [0 4 1]

Performing the Dot Product

Now, we'll perform the dot product of these two vectors. Remember, this means multiplying corresponding elements and then adding the results:

(3 * 0) + (5 * 4) + (-1 * 1) = 0 + 20 - 1 = 19

So, the dot product of the second row of A and the third column of B is 19.

The Result

Therefore, the entry (AB)₂₃ is 19.

  • That's it! We've successfully calculated the entry in the second row and third column of the product matrix AB. See, matrix multiplication isn't so scary after all!

Putting it all Together: The Calculation

To summarize, let's write out the calculation in a more formal way:

(AB)23=[35−1]⋅[041]=(3×0)+(5×4)+(−1×1)=0+20−1=19(AB)_{23} = \begin{bmatrix} 3 & 5 & -1 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 4 \\ 1 \end{bmatrix} = (3 \times 0) + (5 \times 4) + (-1 \times 1) = 0 + 20 - 1 = 19

This clearly shows how we obtained the result of 19.

Why is this Important? Applications of Matrix Multiplication

You might be thinking,