Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right], B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 8 \end{array}\right] $$

Step-by-Step Solution

Answer

The solutions are: (a) \(AB = \left[ \begin{array}{ll} 0 & 15 \ 6 & 8 \end{array} \right]\), (b) \(BA = \left[ \begin{array}{ll} -2 & 2 \ 31 & 12 \end{array} \right]\), (c) \(A^{2} = \left[ \begin{array}{ll} 9 & 6 \ 12 & 8 \end{array} \right]\).

Step 1: Find \(AB\)

First, let's start to find \(AB\) by multiplying matrices A and B together. When multiplying matrices, each element \(a_{ij}\) in the resulting matrix is found by summing the products of elements from the i-th row of the first matrix and j-th column of the second matrix. Therefore, \(AB = A \cdot B\) can be written \[ \left[ \begin{array}{ll} 1 \cdot 2 + 2 \cdot -1 & 1 \cdot -1 + 2 \cdot 8 \ 4 \cdot 2 + 2 \cdot -1 & 4 \cdot -1 + 2 \cdot 8 \end{array} \right] \] which simplifies to \[ \left[ \begin{array}{ll} 0 & 15 \ 6 & 8 \end{array} \right] \]

Step 2: Find \(BA\)

Now let's multiply B and A in the order \(BA = B \cdot A\). Again, we multiply and sum the elements from the i-th row of the first matrix and j-th column of the second matrix for each \(a_{ij}\) in the resulting matrix. Thus, \[ BA = \left[ \begin{array}{ll} 2 \cdot 1 + -1 \cdot 4 & 2 \cdot 2 + -1 \cdot 2 \ -1 \cdot 1 + 8 \cdot 4 & -1 \cdot 2 + 8 \cdot 2 \end{array} \right] \] which simplifies to \[ \left[ \begin{array}{ll} -2 & 2 \ 31 & 12 \end{array} \right] \]. We can see this result is different from that obtained in step 1.

Step 3: Find \(A^2\)

To find \(A^2\), we need to multiply the matrix A by itself. This follows the same steps as in equations (1) and (2) with both matrices being A; i.e., \(A^2 = A \cdot A\). Conducting this operation, we find \[ A^2 = \left[ \begin{array}{ll} 1 \cdot 1 + 2 \cdot 4 & 1 \cdot 2 + 2 \cdot 2 \ 4 \cdot 1 + 2 \cdot 4 & 4 \cdot 2 + 2 \cdot 2 \end{array} \right] = \left[ \begin{array}{ll} 9 & 6 \ 12 & 8 \end{array} \right] \], which is the square of matrix A.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices and operations on them, such as addition, subtraction, and multiplication. In matrix multiplication, we organize numbers into rectangular arrays of rows and columns.
Each number in a matrix is termed an element and the size of a matrix is determined by its number of rows and columns, often represented as m x n where m is the number of rows and n is the number of columns.

• Two matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second.
• The product of matrices A and B, denoted as AB, results in a new matrix where each element is a sum of the products of corresponding elements from rows and columns.

Working with matrix algebra helps simplify complex problems in various fields such as physics, economics, and computer science.

Order of Operations in Matrices

The order of operations is crucial when working with matrices because matrix multiplication is not commutative. This means that, in general, AB is not equal to BA. Therefore, the sequence of multiplication steps matters greatly.

• Always perform operations inside the brackets first if there are any.
• Calculate matrix powers, such as A², by multiplying the matrix by itself.
• Pay attention to the order when multiplying multiple matrices to ensure accurate results.

For instance, in the original exercise, to find AB, the elements of matrix A are multiplied with corresponding elements of matrix B based on their positions in the arrays. Similarly, the resultant matrix is different when calculating BA as compared to AB because of the differing sequence in operations.

Matrix Squaring

Matrix squaring involves multiplying a matrix by itself. This procedure is particularly useful when analyzing systems of equations and transformations. To compute the square of a matrix A, expressed as A², follow these steps:

• Ensure that the matrix is square, meaning the number of rows is equal to the number of columns.
• Multiply each element of the matrix by corresponding elements of the same matrix.
• Sum the product for each element to get the resulting matrix.

The original exercise demonstrates matrix squaring comprehensively. By computing A², or AA, an entirely new matrix is formed, showcasing results unique to this process.
Proper understanding of matrix squaring is essential in various mathematical applications, including linear transformations and describing rotational transformations in geometry.