Problem 24 - Precalculus: Functions and Graphs

Question

Find \(A B\). $$A=\left[\begin{array}{r} 4 \\ -3 \\ 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 5 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The product \(A B\) is \(\left[\begin{array}{cc} 20 & 4 \\ -15 & -3 \\ 10 & 2 \end{array}\right]\).

Step 1: Understand Matrix Dimensions

Matrix \(A\) is a column vector with dimensions \(3 \times 1\), and matrix \(B\) is a row vector with dimensions \(1 \times 2\). Matrix multiplication \(A B\) is possible between these two matrices because the number of columns in \(A\) (which is 1) matches the number of rows in \(B\) (which is 1). The resulting matrix, \(A B\), will have dimensions \(3 \times 2\).

Step 2: Perform Matrix Multiplication

To find the product \(A B\), multiply each element of \(A\) with each element of \(B\) and arrange in a \(3 \times 2\) matrix. The element in the \(i^{th}\) row and \(j^{th}\) column of \(A B\) is obtained by multiplying the \(i^{th}\) element of \(A\) with the \(j^{th}\) element of \(B\).Continue as follows:1. First row of resulting matrix: - \(4 \times 5 = 20\) - \(4 \times 1 = 4\)2. Second row of resulting matrix: - \(-3 \times 5 = -15\) - \(-3 \times 1 = -3\)3. Third row of resulting matrix: - \(2 \times 5 = 10\) - \(2 \times 1 = 2\)

Step 3: Write the Resulting Matrix

After performing the calculations, the elements of the matrix \(A B\) are:\[A B = \left[\begin{array}{cc}20 & 4 \ -15 & -3 \ 10 & 2 \ \end{array}\right]\]

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial for performing matrix operations, especially multiplication. A matrix is defined by the number of its rows and columns. This is commonly represented as \(m \times n\), where \(m\) is the number of rows, and \(n\) is the number of columns.
Consider matrix \(A\), which is given as a \(3 \times 1\) matrix. This means it has 3 rows and 1 column, making it a column vector.
Matrix \(B\) is described as a \(1 \times 2\) matrix, meaning it has 1 row and 2 columns, forming a row vector.

When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The result of multiplying these matrices, \(A B\), will have dimensions based on the outer terms of the dimensions, here \(3 \times 2\). This means the resulting matrix will have 3 rows and 2 columns.

Column Vector

A column vector is a special type of matrix with only one column. It's essentially a vertical list of elements.
For example, matrix \(A\) is a column vector:

• The dimensions are \(3 \times 1\).
• Each element, from top to bottom, is 4, -3, and 2.

This structure makes a column vector useful in linear algebra for representing points in a space or specific quantities. Column vectors are pivotal when performing certain operations, such as dot products and forming linear combinations.

Row Vector

A row vector is another specific type of matrix but with only one row and multiple columns, creating a horizontal list of elements. Matrix \(B\) is an example of a row vector:

• The dimensions are \(1 \times 2\).
• The elements are arranged horizontally as 5 and 1.

Row vectors are powerful tools in linear algebra for representing equations, transformations, or compiling data in a concise way. Like column vectors, they play a vital role in various mathematical computations, including matrix multiplication.

Resulting Matrix

The resulting matrix from multiplying two matrices depends on the compatibility of their dimensions. The process involves multiplying corresponding elements and summing up the products.
When multiplying the given matrices \(A\) and \(B\):

• The first element of \(A B\) is obtained by multiplying the first element of \(A\) with each element of \(B\) (i.e., \(4 \times 5 = 20\) and \(4 \times 1 = 4\)).
• The second element follows the same pattern with the second row of \(A\) and elements of \(B\) (i.e., \(-3 \times 5 = -15\) and \(-3 \times 1 = -3\)).
• This pattern continues for each row of \(A\).

After completing the multiplications and arranging them in sequence, you end up with the resulting matrix: \\[ \left[ \begin{array}{cc} 20 & 4 \ -15 & -3 \ 10 & 2 \end{array} \right] \]This is the \(3 \times 2\) output we anticipated from the multiplication rules, matching our dimension analysis.