Problem 34 - Precalculus with Limits

Question

In Exercises 33-40, if possible, find \(AB\) and state the order of the result. \(A=\left[\begin{array}{r} 0 & -1 & 2 \\ 6 & 0 & 3 \\ 7 & -1 & 8 \end{array}\right]\), \(B=\left[\begin{array}{r} 2 & -1 \\ 4 & -5 \\ 1 & 6 \end{array}\right]\)

Step-by-Step Solution

Answer

Matrix AB is a 3x2 matrix and its elements are \[AB=\left[\begin{array}{r} -2 & 17 \ 24 & 24 \ 26 & 37 \end{array}\right]\]

Step 1: Analyze the matrices

First, consider the given matrices. Matrix A is a 3x3 matrix because it has 3 rows and 3 columns. Matrix B is a 3x2 matrix because it has 3 rows and 2 columns. The number of columns in matrix A equals the number of rows in matrix B, which means we can perform the multiplication.

Step 2: Multiply the matrices

When multiplying matrices, elements in the resultant matrix are generated by taking the dot product of corresponding rows from the first matrix and columns from the second matrix. For instance, the (1,1) position of Matrix AB will be computed as (0*2)+(-1*4)+(2*1) = -2, the (1,2) position as (0*-1)+(-1*-5)+(2*6) = 17, and so forth for all other elements of the resultant matrix

Step 3: Obtain the Matrix AB and state its order

Conduct similar operations for all the elements of the matrices to obtain Matrix AB. The order of the resultant matrix AB will be the number of rows from matrix A by the number of columns from matrix B, thus the order of the resultant matrix is 3x2

Key Concepts

Understanding Matrix Order

Matrix order is a way to describe the dimensions of a matrix. It is expressed as the number of rows by the number of columns. For example, a 3x3 matrix means there are 3 rows and 3 columns.

It's essential to understand the order because it dictates if two matrices can be multiplied. Specifically, the number of columns in the first matrix must match the number of rows in the second matrix. In our example, Matrix A is 3x3 and Matrix B is 3x2.

This order compatibility allows the matrices to be multiplied, resulting in a new matrix. This new matrix will have the order defined by the number of rows from Matrix A and the number of columns from Matrix B, making it a 3x2 matrix.

The Dot Product in Matrix Multiplication

The dot product is a foundational operation in matrix multiplication. It involves multiplying corresponding elements of a row from one matrix by the elements of a column in the other matrix and then summing those products.

For example, to find the element at the (1,1) position within the product matrix AB, you'd perform the following calculation:

• Multiply each element in the first row of Matrix A by the corresponding elements in the first column of Matrix B.
• \( (0 \times 2) + (-1 \times 4) + (2 \times 1) = -2 \)

The resulting value, -2 in this case, becomes the entry in the first row and first column of the resultant matrix.

This process repeats for each position in the resultant matrix by using corresponding rows and columns from the original matrices.

Building the Resultant Matrix

After performing the dot product operations for each position in the resultant matrix, you construct the new matrix. This matrix, formed after multiplying two matrices, is called the resultant matrix.

For our specific case:

• The resultant matrix AB is 3x2 because Matrix A has 3 rows and Matrix B has 2 columns.
• Each element in this 3x2 matrix is determined through the dot product of relevant rows and columns from both matrices.

This systematic approach allows us to build the resultant matrix efficiently. Its order reflects the dimensions dictated by the original matrices' compatible rows and columns.