Problem 27 - Algebra and Trigonometry

Question

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{ll} {1} & {3} \\ {5} & {3} \end{array}\right], \quad B=\left[\begin{array}{rr} {3} & {-2} \\ {-1} & {6} \end{array}\right] $$

Step-by-Step Solution

Answer

The resultant matrices are \(AB=\left[\begin{array}{rr}{0} & {16}\ {12} & {8}\end{array}\right]\) and \(BA=\left[\begin{array}{rl}{-7} & {3}\ {27} & {12}\end{array}\right]\).

Step 1: Multiplying Matrix A by Matrix B

To find AB, each element in the resultant matrix is calculated by multiplying corresponding elements and summing. For example, element (1,1) in AB is computed as follows: \(a_{11} = 1*3 + 3*-1 = 0\). Continue this process for each element. The resultant matrix after all this computation is referred to as \(AB\).

Step 2: Compute Matrix A times Matrix B

Now we find \(AB = \left[\begin{array}{ll}{1} & {3} \ {5} & {3}\end{array}\right] \times \left[\begin{array}{rr} {3} & {-2} \ {-1} & {6}\end{array}\right]\). This gives us: \(AB=\left[\begin{array}{rr}{0} & {16}\ {12} & {8}\end{array}\right]\).

Step 3: Multiplying Matrix B by Matrix A

Just like in step 1, we will take the dot product of rows of Matrix B with the corresponding Columns of Matrix A to get the product BA. Again, each element in the resultant matrix is calculated by multiplying each element in the corresponding row of the first matrix by each element in the corresponding column of the second matrix and summing.

Step 4: Compute Matrix B times Matrix A

Finally we find \(BA = \left[\begin{array}{rr} {3} & {-2} \ {-1} & {6}\end{array}\right] \times \left[\begin{array}{ll}{1} & {3} \ {5} & {3}\end{array}\right]\). This yields: \(BA=\left[\begin{array}{rl}{-7} & {3}\ {27} & {12}\end{array}\right]\).

Key Concepts

Matrices

Matrices are rectangular arrays composed of numbers placed in rows and columns. They are fundamental tools in various fields such as mathematics, computer science, and engineering. The size or dimension of a matrix is defined by the number of its rows and columns. For instance, if a matrix has 2 rows and 3 columns, it is called a 2x3 (read as "2 by 3") matrix.

Each element within a matrix is identified by a pair of indices, indicating its row and column position. These are usually denoted as (i,j), where 'i' is the row number and 'j' is the column number. Matrices can be used to store data, solve systems of linear equations, and even transform graphical data in computer games.

• Matrix A is a 2x2 matrix shown as \( \begin{bmatrix} 1 & 3 \ 5 & 3 \end{bmatrix} \).
• Matrix B is another 2x2 matrix represented by \( \begin{bmatrix} 3 & -2 \ -1 & 6 \end{bmatrix} \).

Matrix Product

The matrix product, also known as matrix multiplication, is a binary operation that takes a pair of matrices and produces another matrix. Understanding how to multiply matrices is critical for tasks such as solving linear systems, transforming coordinates, and changing data dimensions.

When multiplying matrices, it's important to ensure that the number of columns in the first matrix equals the number of rows in the second matrix. This is crucial for the operation to be defined. In the exercise given, both matrices A and B are 2x2 matrices, making them compatible for multiplication in both orders, AB and BA.

• The element in the first row, first column of AB is calculated by combining the products of corresponding elements: \(1 \times 3 + 3 \times -1 = 0\).
• Similarly, to find an element like the second row, first column of BA, perform: \(3 \times 1 + -2 \times 5 = -7\).

Thus, multiplying matrices requires careful attention to their arrangement and compatibility.

Matrix Operations

Matrix operations involve various ways to manipulate matrices to achieve desired outcomes. Besides multiplication, common matrix operations include addition and subtraction. However, multiplication differs from addition and subtraction in terms of rules and applications.

One significant aspect of matrix multiplication is that it is not commutative. This means that generally, the matrix product AB is not the same as BA, as illustrated in our exercise.

• From our calculations, \(AB = \begin{bmatrix} 0 & 16 \ 12 & 8 \end{bmatrix} \)
• Whereas \(BA = \begin{bmatrix} -7 & 3 \ 27 & 12 \end{bmatrix} \)

This highlights the importance of the order in which matrices are multiplied. Apart from multiplication, transpose and determinant are among other common operations that provide valuable insights into matrix properties. Understanding these operations are key skills when working with matrices in various contexts.