Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Find \(A B\), if possible. $$ A=\left[\begin{array}{r} 6 \\ -2 \\ 1 \\ 6 \end{array}\right], B=\left[\begin{array}{ll} 10 & 12 \end{array}\right] $$

Step-by-Step Solution

Answer

The product AB is \( \[ \begin{matrix} 60 & 72 \ -20 & -24 \ 10 & 12 \ 60 & 72 \end{matrix} \]\)

Step 1: Setup the multiplication

We write the matrix multiplication AB on the paper or screen: \( A B=\left[\begin{array}{r} 6 \ -2 \ 1 \ 6 \end{array}\right] \left[\begin{array}{ll} 10 & 12 \end{array}\right] \). Now our job is to perform this multiplication.

Step 2: Perform the multiplication

Here each element of the 4x1 matrix (A) is multiplied with each element of 1x2 matrix (B) respectively. \So, we get \\( 6(10)\),\( 6(12)\),\\(-2(10)\),\(-2(12)\),\\(1(10)\), \(1(12)\),\\(6(10)\), \(6(12)\),\which upon calculating simplifies to \60, 72,\-20, -24,\10, 12,\60, 72.

Step 3: Write the product

We now arrange these products into a 4x2 matrix (the dimensions of our resulting matrix from Step 1), making sure to keep the order consistent: \( A B = \[ \begin{matrix} 60 & 72 \ -20 & -24 \ 10 & 12 \ 60 & 72 \end{matrix} \]\)

Key Concepts

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It provides a formal structure for many natural and social sciences, including physics, computer science, economics, and engineering. In linear algebra, the concept of a matrix is central, as it's a compact way to represent and operate with linear transformations or systems.

Matrix operations, such as matrix multiplication, help in solving linear equations and can describe geometric transformations. Understanding the foundation of linear algebra thus becomes crucial for disciplines that rely heavily on the manipulation and interpretation of multi-dimensional data.

Matrices

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The size of a matrix is defined by the number of its rows and columns and is denoted as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.

Each element in a matrix has a specific position, identified by its row and column number. For instance, in a 3x3 matrix, the top left element is the first element of the first row and the first column. Matrices are used to represent and solve systems of linear equations and to perform various operations on data, which are foundational elements in linear algebra.

Matrix Operations

Matrix operations include addition, subtraction, scalar multiplication, and matrix multiplication. Each operation has its own set of rules and properties.

Matrix Multiplication

It is perhaps the most interesting, as it involves a rule that is not immediately intuitive. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The result of the multiplication is a new matrix with the number of rows of the first matrix and the number of columns of the second matrix.

When we multiply each row of the first matrix with each column of the second, the numbers are multiplied in pairs and then added together to form the entries of the product matrix. This process can be visualized with the exercise provided, where a 4x1 matrix multiplies a 1x2 matrix resulting in a 4x2 product matrix. Each element of the first matrix multiplies with each element of the second matrix, which is an excellent demonstration of the procedure.