Problem 11 - Calculus for Biology and Medicine

Question

$$ \begin{array}{r} A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ 2 & 4 & 1 \\ 0 & -2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 5 & -1 & 4 \\ 2 & 0 & 1 \\ 1 & -3 & -3 \end{array}\right], \\ \quad C=\left[\begin{array}{rrr} -2 & 0 & 4 \\ 1 & -3 & 1 \\ 0 & 0 & 2 \end{array}\right] \end{array} $$ $$ \text { Show that } A+B=B+A \text { . } $$

Step-by-Step Solution

Answer

Matrix addition is commutative: \(A + B = B + A\).

Step 1: Write the Expression

We need to verify that matrix addition is commutative for matrices \(A\) and \(B\). To do this, calculate \(A + B\) and \(B + A\), and then compare them.

Step 2: Calculate A + B

Add corresponding elements of matrices \(A\) and \(B\):\[A + B = \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} + \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} = \begin{bmatrix} 1+5 & 3-1 & -1+4 \ 2+2 & 4+0 & 1+1 \ 0+1 & -2-3 & 2-3 \end{bmatrix} = \begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix}\]

Step 3: Calculate B + A

Add corresponding elements of matrices \(B\) and \(A\) (note the order is reversed):\[B + A = \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} + \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} = \begin{bmatrix} 5+1 & -1+3 & 4-1 \ 2+2 & 0+4 & 1+1 \ 1+0 & -3-2 & -3+2 \end{bmatrix} = \begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix}\]

Step 4: Compare A + B and B + A

Notice that the result for both \(A + B\) and \(B + A\) is the same. They both result in \(\begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix}\).

Step 5: Conclusion

Since \(A + B = B + A\), we have shown that the matrix addition is commutative for the given matrices \(A\) and \(B\).

Key Concepts

Commutative Property

When learning about matrix addition, it's crucial to understand the commutative property. In simple terms, this means that the order in which you add the matrices does not matter. In algebra, this is familiar with numbers where for instance, 2 + 3 is the same as 3 + 2. The same principle applies to matrices.

For matrices of the same dimensions, if you have an operation like adding matrices \(A\) and \(B\), the commutative property tells us that \(A + B\) is the same as \(B + A\). The exercise demonstrated this property by calculating and comparing the matrices:

• Calculate \(A + B\) by adding each corresponding element.
• Calculate \(B + A\) in the same fashion.
• Verify that both results are identical.

Recognizing the commutative property in matrix addition streamlines solving larger matrix problems and ensures correct addition regardless of matrix order.

Matrix Operations

Matrix operations involve several key actions like addition, subtraction, and multiplication of matrices. In the context of this exercise, we're focusing on matrix addition.

When adding matrices, it's important to note:

• Matrices must be of the same size (same number of rows and columns) to perform addition.
• Add each corresponding element from each matrix. That means the element in the first row and first column of matrix \(A\) will be added to the element in the first row and first column of matrix \(B\), and so forth.

This requires careful alignment and attention to the individual positions of elements within the matrices.
For example, in this exercise, when calculating \(A + B\), you added the elements at respective positions, such as \(1 + 5\), \(3 + (-1)\), etc.

Understanding this foundational operation is crucial for all more complex matrix manipulations you might encounter later on.

Matrix Arithmetic

Matrix arithmetic encompasses a variety of operations that can be performed on matrices. Beyond addition, matrix arithmetic includes subtraction and multiplication. However, each has specific rules that govern their processes.

In matrix addition, the concept is fairly straightforward:

• The matrices must be of the same dimension.
• Each element of one matrix is added to the corresponding element of the other matrix.

When working with matrices:

• For subtraction, it’s similar to addition but you subtract each corresponding element instead.
• For multiplication, a different rule applies which involves the dot product of rows and columns. This requires specific alignment and often different dimensions between matrices.

Matrix arithmetic allows for extensive manipulation of data commonly used in computer science, engineering, and applied mathematics fields. Becoming comfortable with these rules aids in understanding complex systems modeled by matrices.