Problem 39 - Precalculus with Limits

Question

In Exercises 33-40, if possible, find \(AB\) and state the order of the result. \(A=\left[\begin{array}{r} 10 \\ 12 \end{array}\right]\), \(B=\left[\begin{array}{r} 6 & -2 & 1 & 6 \end{array}\right]\)

Step-by-Step Solution

Answer

The product of \(A\) and \(B\) is \(\left[\begin{array}{cccc} 60 & -20 & 10 & 60 \ 72 & -24 & 12 & 72 \end{array}\right]\) which is a 2x4 matrix.

Step 1: Set Up the Matrices

Matrix \(A\) has two rows and one column, that contains the elements [10, 12]. Matrix \(B\) has one row and four columns, that contains the elements [6, -2, 1, 6].

Step 2: Multiply the Matrices

The product of \(A\) and \(B\) is calculated by multiplying elements and adding up. First, we multiply the first row of \(A\) by the first column of \(B\), and then add the products. Repeat this for all rows in \(A\) and columns in \(B\).

Step 3: Calculate the Elements of the New Matrix

The first element is \(10*6=60\), second element is \(10*(-2)=-20\), third element is \(10*1=10\) and forth is \(10*6=60\). Repeat this with the second row of \(A\), so the fifth element is \(12*6=72\), sixth element is \(12*(-2)=-24\), seventh element is \(12*1=12\) and the last element is \(12*6=72\).

Key Concepts

Understanding Matrix Order

When we talk about matrices, understanding their order is crucial. The order of a matrix is determined by its dimensions, which are specified as "rows by columns." This directly influences how matrix multiplication is performed and whether it is even possible.

• Matrix A is given as \(2 \times 1\), meaning it has 2 rows and 1 column.
• Matrix B is \(1 \times 4\), with 1 row and 4 columns.
• For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, since Matrix A has 1 column and Matrix B has 1 row, they satisfy the condition.

The resulting matrix will have the dimensions of the outside numbers: \(2 \times 4\). Therefore, the product of \(AB\) is possible and will result in a matrix with 2 rows and 4 columns.

Matrix Elements

Matrix elements are the individual numbers within a matrix. Each element is uniquely positioned based on its row and column index, which is crucial when performing operations like matrix multiplication.

• Matrix A consists of elements [10, 12], positioned in two separate rows.
• Matrix B has elements [6, -2, 1, 6], all belonging to a single row.

When multiplying matrices, you take each element from the row of the first matrix and multiply it with the corresponding element from the column of the second matrix. The results are then added together to form the elements of the new matrix. Understanding this will help in visualizing and calculating matrix products accurately.

The Matrix Product

The matrix product is the result of multiplying two matrices. This process might seem complex initially, but it's simplified by following systematic steps.
For matrices \(A\) and \(B\), we:

• Multiply each element of the row in Matrix A with each element of the column in Matrix B.
• Add these products to get a single element in the resulting matrix.
• Repeat this process for each row of A and column of B.

In this particular exercise:

• The first element: \(10 \times 6 = 60\), and similarly, for the column elements, yielding the output row [60, -20, 10, 60].
• Using the second row: \(12 \times 6 = 72\), producing the row [72, -24, 12, 72].

Thus, the matrix product \(AB\) results in a new \(2 \times 4\) matrix:\[\begin{array}{cccc}60 & -20 & 10 & 60 \72 & -24 & 12 & 72 \\end{array}\]