Problem 23 - Calculus for Biology and Medicine

Question

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Show that } A C \neq C A \text { . } $$

Step-by-Step Solution

Answer

Matrix multiplication is not commutative: \(AC \neq CA\).

Step 1: Understand Matrix Multiplication

Matrix multiplication is not commutative, which means that the order in which you multiply matrices matters. To show that two matrices are not commutative, we need to calculate their product in both orders, i.e., multiply matrix \(A\) by matrix \(C\) and then \(C\) by \(A\), and demonstrate that these two products are not equal.

Step 2: Calculate Product AC

Compute the matrix product \(AC\). Given \(A = \begin{bmatrix} -1 & 0 \ 1 & 2 \end{bmatrix}\) and \(C = \begin{bmatrix} 1 & 2 \ 0 & -1 \end{bmatrix}\), calculate:\[AC = \begin{bmatrix} -1 & 0 \ 1 & 2 \end{bmatrix} \begin{bmatrix} 1 & 2 \ 0 & -1 \end{bmatrix} = \begin{bmatrix} (-1)(1) + (0)(0) & (-1)(2) + (0)(-1) \ (1)(1) + (2)(0) & (1)(2) + (2)(-1) \end{bmatrix} = \begin{bmatrix} -1 & -2 \ 1 & 0 \end{bmatrix}\]

Step 3: Calculate Product CA

Compute the matrix product \(CA\). Given \(C = \begin{bmatrix} 1 & 2 \ 0 & -1 \end{bmatrix}\) and \(A = \begin{bmatrix} -1 & 0 \ 1 & 2 \end{bmatrix}\), calculate:\[CA = \begin{bmatrix} 1 & 2 \ 0 & -1 \end{bmatrix} \begin{bmatrix} -1 & 0 \ 1 & 2 \end{bmatrix} = \begin{bmatrix} (1)(-1) + (2)(1) & (1)(0) + (2)(2) \ (0)(-1) + (-1)(1) & (0)(0) + (-1)(2) \end{bmatrix} = \begin{bmatrix} 1 & 4 \ -1 & -2 \end{bmatrix}\]

Step 4: Compare AC and CA

Compare the results from Step 2 and Step 3: \[AC = \begin{bmatrix} -1 & -2 \ 1 & 0 \end{bmatrix}\] and \[CA = \begin{bmatrix} 1 & 4 \ -1 & -2 \end{bmatrix}\]. Clearly, \(AC eq CA\).

Key Concepts

Non-Commutative Property

Matrix multiplication has a peculiar property known as the non-commutative property. Simply put, this means that when you multiply two matrices, the order in which you multiply them affects the result. In other words, for matrices \(A\) and \(B\), \(AB\) is generally not equal to \(BA\).

This is different from basic arithmetic. For example, in basic numbers, 2 multiplied by 3 is the same as 3 multiplied by 2. However, for matrices, this symmetry doesn't apply.

To see this non-commutative nature in action, one can consider multiplying matrices \(A\) and \(C\) in two different orders: first as \(AC\) and then as \(CA\), show that their products are not equal. This disparity highlights the importance of recognizing the non-commutative property when working with matrix products.

Matrix Product

The matrix product involves a specific technique of multiplying the rows of the first matrix by the columns of the second. To correctly perform matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

For example, if matrix \(A\) is a 2x2 matrix and matrix \(C\) is also a 2x2 matrix, we can multiply them. When doing so:

• The result will also be a 2x2 matrix because each dimension between the two matrices is consistent.
• Each element of the resulting matrix is derived by multiplying corresponding elements and summing their products.

Consider when multiplying the first row of \(A\) by the first column of \(C\), we'd compute each paired element, add them together, and repeat this process across the matrix to populate the corresponding positions in the resulting matrix.

Performing multiplication in matrix form often leads to quite different results depending on the multiplication order, emphasizing its non-commutative nature. This ensures a refreshed approach is required compared to standard numeric multiplication.

Calculation Steps

When calculating matrix products, such as \(AC\) and \(CA\), following structured steps will help ensure accuracy and understanding:

1. **Set Up the Matrices:** Start by clearly laying out the matrices you want to multiply. Ensure the number of columns in the first matches the rows in the second.
2. **Multiply and Sum:** Each element of the new matrix is calculated by multiplying corresponding elements of the specified row from the first matrix by the specified column from the second. Sum these products for each position in the resulting matrix. For example:

• For \(AC\), compute as follows: \((-1\times1 + 0\times0, -1\times2 + 0\times{-1})\) for the first row.
• For \(CA\), calculate similarly: \((1\times{-1} + 2\times1, 1\times0 + 2\times2)\).

3. **Compare Results:** Carefully compare the resulting matrices from each multiplication order to verify differences. As shown, different orders often yield different results, affirming the non-commutative property.

Follow these steps diligently to master the technique of calculating matrix products, ensuring a deeper understanding of the intricacies of matrix algebra.