Problem 10 - Contemporary Precalculus

Question

Find AB. $$A=\left(\begin{array}{rrr} -1 & 7 & 1 \\ -5 & 3 & 2 \\ 0 & 1 & 5 \\ -3 & 6 & 7 \end{array}\right), \quad B=\left(\begin{array}{rrrr} 7 & -2 & 6 & 2 \\ -2 & 8 & 4 & 1 \\ 0 & 7 & 0 & -5 \end{array}\right)$$

Step-by-Step Solution

Answer

Answer: The resulting product matrix AB is: $$AB = \left(\begin{array}{rrrr} -21 & 65 & -8 & -39 \\ -31 & -46 & -2 & 15 \\ 0 & 7 & 0 & -5 \\ -14 & 58 & 6 & -12 \end{array}\right)$$

Step 1: Determine the dimensions of matrices A and B

Matrix A has 4 rows and 3 columns, so its dimensions are 4x3. Matrix B has 3 rows and 4 columns, hence its dimensions are 3x4.

Step 2: Check if multiplication is possible

In order for matrix multiplication to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Since matrix A has 3 columns and matrix B has 3 rows, we can multiply them.

Step 3: Perform the matrix multiplication

To multiply the matrices, we will perform the row-by-column method. The resulting matrix, AB, will have dimensions 4x4. For element (1,1) in the product matrix: $$AB_{11} = (-1)(7) + (7)(-2) + (1)(0) = -7-14+0 = -21$$ For element (1,2) in the product matrix: $$AB_{12} = (-1)(-2) + (7)(8) + (1)(7) = 2+56+7 = 65$$ Continue doing this for all the elements in the resulting matrix AB: $$AB = \left(\begin{array}{rrrr} -21 & 65 & -8 & -39 \\ -31 & -46 & -2 & 15 \\ 0 & 7 & 0 & -5 \\ -14 & 58 & 6 & -12 \end{array}\right)$$

Key Concepts

Matrices Dimensions

Understanding the dimensions of matrices is crucial before attempting any matrix operations. The dimension of a matrix is expressed as the number of rows by the number of columns it contains. For example, if a matrix has 4 rows and 3 columns, its dimension is denoted as a 4x3 matrix.

Knowing the dimensions allows you to determine if certain operations, like multiplication, can be performed. The rule for matrix multiplication is straightforward: the number of columns in the first matrix must equal the number of rows in the second matrix. Therefore, a 4x3 matrix can be multiplied with a 3x4 matrix, resulting in a 4x4 matrix. This rule is not just a mathematical formality; it ensures that the operation is computationally feasible.

Row-by-Column Method

The row-by-column method is the standard procedure for multiplying two matrices. To multiply, you take the elements of each row of the first matrix and multiply them correspondingly with the elements of each column in the second matrix, then sum those products to get a single element in the resulting matrix.

Let's break it down further: to find the element located in the first row and first column of the product matrix, multiply the elements of the first row of the first matrix by the elements of the first column of the second matrix. Repeat this for each element by working across the rows and down the columns. This is a repetitive process that can be time-consuming, but it ensures the correct computation of each element in the product matrix.

Matrix Operations

Matrix operations include addition, subtraction, and multiplication. However, each operation follows a unique set of rules that must be strictly adhered to. For multiplication, the row-by-column method is used, where we systematically compute the elements of the resulting matrix. Other operations, like addition and subtraction, are simpler as they only require element-by-element arithmetic, but require the matrices to be the same dimension.

It’s important to note that matrix multiplication is not commutative; this means that the order in which you multiply the matrices matters. Multiplying matrix A by matrix B (AB) generally yields a different result than multiplying matrix B by matrix A (BA). Each operation plays a fundamental role in various mathematical and application fields, including linear algebra, computer graphics, and systems of equations.

Precalculus

Precalculus is a course that prepares students for calculus and other higher-level mathematics. It covers a wide array of topics, including algebra, functions, complex numbers, and notably, matrix operations. It represents a bridge between algebraic concepts and the more advanced world of calculus. Matrix multiplication, particularly, is a precalculus topic that has vast applications in calculus, like dealing with linear transformations and systems of linear equations.

Understanding matrix operations thoroughly in precalculus is essential. These concepts are not just abstract mathematical ideas; they are practical tools for solving real-world problems in physics, engineering, economics, and more. Therefore, a strong grasp of these precalculus topics can solidify the foundational skills necessary for success in many STEM fields.