Problem 21 - Calculus for Biology and Medicine

Question

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \begin{array}{l} \text { Compute the following: }\\\ A B \quad \text { (b) } \underline{B A} \end{array} $$

Step-by-Step Solution

Answer

(a) \( AB = \begin{bmatrix} -2 & -3 \\ 0 & 5 \end{bmatrix} \); (b) \( BA = \begin{bmatrix} 1 & 6 \\ 2 & 2 \end{bmatrix} \).

Step 1: Understand the Task

We need to perform matrix multiplication for two expressions: (a) \( AB \) and (b) \( BA \), given matrices \( A \) and \( B \). Matrix multiplication is not commutative, so \( AB \) may not equal \( BA \).

Step 2: Multiply Matrices A and B

For \( AB \), we multiply matrix \( A \) by matrix \( B \). Using the rule of multiplying matrices, the element in the first row and first column of \( AB \) is \((-1)\times 2 + 0\times(-1) = -2\). The element in the first row and second column is \((-1)\times 3 + 0\times 1 = -3\). In the second row, the first column element is \(1\times 2 + 2\times(-1) = 2 - 2 = 0\), and the second row, second column element is \(1\times 3 + 2\times 1 = 3 + 2 = 5\). Therefore, \( AB = \begin{bmatrix} -2 & -3 \ 0 & 5 \end{bmatrix} \).

Step 3: Multiply Matrices B and A

For \( BA \), we multiply matrix \( B \) by matrix \( A \). The first row and first column element is \(2\times(-1) + 3\times 1 = -2 + 3 = 1\). The first row and second column element is \(2\times 0 + 3\times 2 = 6\). The second row, first column element is \((-1)\times(-1) + 1\times 1 = 1 + 1 = 2\), and the second row, second column element is \((-1)\times 0 + 1\times 2 = 2\). Therefore, \( BA = \begin{bmatrix} 1 & 6 \ 2 & 2 \end{bmatrix} \).

Key Concepts

Commutative Property

The commutative property is a mathematical principle frequently encountered in basic arithmetic, particularly with addition and multiplication. It states that the order of numbers in an operation does not change the result. For example, in addition, the commutative property tells us that \( a + b = b + a \), and for multiplication, \( a \times b = b \times a \). However, when dealing with matrices, it's crucial to understand that matrix multiplication does not follow this property. That means for two matrices \( A \) and \( B \), it is generally not true that \( AB = BA \).

In our original exercise, we computed both \( AB \) and \( BA \) to see that these products are different matrices, exemplifying the non-commutative nature of matrix multiplication. Therefore, while you might expect operations with numbers to be interchangeable, matrices require careful attention to order, as swapping positions alters the outcome.

Matrices

Matrices are rectangular arrays of numbers and are fundamental in various fields of science, especially in mathematics, physics, and computer science. Each matrix is defined by its number of rows and columns, known as its dimensions. For instance, a 2x2 matrix has two rows and two columns. Matrices are used to represent linear transformations and perform operations such as addition, subtraction, and multiplication.

In our exercise, matrices \( A \) and \( B \) are both 2x2 matrices. This is important because only matrices of compatible dimensions can be multiplied. Specifically, if \( A \) is an \( m \times n \) matrix, and \( B \) is an \( n \times p \) matrix, then their product \( AB \) will be an \( m \times p \) matrix. Since both \( A \) and \( B \) here are 2x2, they can indeed be multiplied together. Understanding how matrices are structured will help in comprehending their manipulation.

Step-by-step Solution

Breaking down matrix multiplication into step-by-step solutions can be very helpful in understanding how the result is calculated. Here is how you resolve such problems efficiently:

• First, identify the matrices you need to multiply. Here, we work with \( A \) and \( B \).
• Next, write out the formula for matrix multiplication. Multiplying two matrices involves taking the dot product of rows from the first matrix with columns of the second matrix.
• Perform the multiplication for each element in the resulting matrix. For example, to find the element in the first row and first column of the resultant matrix, you multiply the corresponding elements in the row of the first matrix with the column of the second matrix and then sum these products.

In our solution, for \( AB \), we calculated each element step by step:

• The element in the first row, first column was calculated as \((-1 \times 2) + (0 \times -1) = -2 \).
• Similarly, compute other elements methodically.

This systematic approach ensures each product is accurate and helps you understand the flow from left to right, top to bottom when handling matrix multiplication.