Problem 35 - Precalculus with Limits

Question

In Exercises 33-40, if possible, find \(AB\) and state the order of the result. \(A=\left[\begin{array}{r} -1 & 6 \\ -4 & 5 \\ 0 & 3 \end{array}\right]\), \(B=\left[\begin{array}{r} 2 & 3 \\ 0 & 9 \end{array}\right]\)

Step-by-Step Solution

Answer

The product of matrices A and B, \(AB\), is \(\left[\begin{array}{cc} -2 & 51 \ -8 & 39 \ 0 & 27 \end{array}\right]\) and the order of the result is 3x2.

Step 1: Confirm if multiplication is possible

Check if the number of columns in matrix A is equal to the number of rows in matrix B. A is a 3x2 matrix and B is a 2x2 matrix. Because both have 2 in common (2 columns for A and 2 rows for B), multiplication is indeed possible.

Step 2: Multiply the matrices

Each element in the resulting matrix is calculated by multiplying the elements in the corresponding row of the first matrix by the elements in the corresponding column of the second matrix and adding the results. Thus, \(AB\) becomes: \[AB = A \times B = \begin{bmatrix} -1\times2 + 6\times0 & -1\times3 + 6\times9 \ -4\times2 + 5\times0 & -4\times3 + 5\times9 \ 0\times2 + 3\times0 & 0\times3 + 3\times9 \end{bmatrix}=\begin{bmatrix} -2 & 51 \ -8 & 39 \ 0 & 27 \end{bmatrix}\]

Step 3: State the order of the result

The order of the resulting matrix is found by counting the number of rows and columns. In this case, there are 3 rows and 2 columns, therefore the order of \(AB\) is 3x2.

Key Concepts

Matrices

A matrix is an array of numbers arranged in rows and columns that is used in mathematical computations. Matricies can represent systems of linear equations, transformations, and data in a compact form, making them a cornerstone concept in precalculus and many applications in science and engineering.

For example, in the context of the exercise provided, matrices A and B are used to perform matrix multiplication, which is a means of combining these structured arrays to derive a new matrix that is a reflection of the compound linear transformation they represent.

Order of a Matrix

The 'order' of a matrix is essentially its size, defined by the number of rows and columns it contains. It's denoted as 'm x n', where 'm' is the number of rows and 'n' is the number of columns. Understanding the order is crucial when performing operations such as multiplication, as it determines the compatibility of matrices.

As illustrated in the solution, matrix A has an order of 3x2 (3 rows and 2 columns), and matrix B has an order of 2x2, which implies that matrix A can be multiplied by matrix B because the number of columns in A matches the number of rows in B. The resulting matrix AB will have an order that reflects the outer dimensions of the matrices, in this case, 3x2.

Arithmetic Operations on Matrices

Just like with numbers, you can perform arithmetic operations with matrices, including addition, subtraction, and multiplication; each operation follows specific rules. Notably, matrices must be of the same order to be added or subtracted. However, for multiplication, the key is to match the number of columns in the first matrix with the number of rows in the second matrix.

For example, when multiplying matrix A and matrix B, as in the original exercise, we take each element of the first row of A multiplied by the corresponding element of the first column of B and add them together to get the element in the first row, first column of the resulting matrix. This process is repeated for every element of the resultant matrix.

Precalculus

Precalculus is an area of mathematics that prepares students for calculus, covering a plethora of topics including functions, complex numbers, and yes, matrices. In precalculus, students learn how to handle various types of functions and transformations, anticipate their graphical representations, and solve real-world problems algebraically.

Matrix multiplication, as encountered in the exercise, is a fundamental precalculus topic, as it helps in understanding linear transformations and systems of equations — key concepts for anyone moving towards calculus. Precalculus serves as the bridge that helps students cross over from the world of basic algebra into the richer realm of calculus.