Problem 11 - Calculus for Biology and Medicine

Question

Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mathbf{4} \\\ \mathbf{- 2} & \mathbf{0} & \mathbf{- 1} \\ \mathbf{1} & \mathbf{3} & \mathbf{3}\end{array}\right] ,\boldsymbol{C}=\left[\begin{array}{lll}\mathbf{1} & \mathbf{0} & \mathbf{4} \\\ \mathbf{0} & \mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{0} & \mathbf{2}\end{array}\right]$$ Show that \(A+B=B+A\).

Step-by-Step Solution

Answer

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

Step 1: Define Matrix Addition

Matrix addition involves adding corresponding elements from the two matrices. For two matrices \(X\) and \(Y\) of the same dimension, their sum \(X+Y\) results in a matrix where each element \((i,j)\) is given by \((X_{ij} + Y_{ij})\).

Step 2: Add Matrices A and B

Begin by adding corresponding elements of matrices \(A\) and \(B\).\[A + B = \begin{bmatrix}1 + 1 & 0 - 1 & 1 + 4 \2 - 2 & 3 + 0 & -1 - 1 \0 + 1 & -2 + 3 & 0 + 3\end{bmatrix} = \begin{bmatrix}2 & -1 & 5 \0 & 3 & -2 \1 & 1 & 3\end{bmatrix}\]

Step 3: Add Matrices B and A

Next, add the corresponding elements of matrices \(B\) and \(A\).\[B + A = \begin{bmatrix}1 + 1 & -1 + 0 & 4 + 1 \-2 + 2 & 0 + 3 & -1 - 1 \1 + 0 & 3 - 2 & 3 + 0\end{bmatrix} = \begin{bmatrix}2 & -1 & 5 \0 & 3 & -2 \1 & 1 & 3\end{bmatrix}\]

Step 4: Compare Results

Observe that both results from \(A + B\) and \(B + A\) are identical.\[A + B = \begin{bmatrix}2 & -1 & 5 \0 & 3 & -2 \1 & 1 & 3\end{bmatrix}\]\[B + A = \begin{bmatrix}2 & -1 & 5 \0 & 3 & -2 \1 & 1 & 3\end{bmatrix}\]This demonstrates that \(A + B = B + A\).

Key Concepts

Commutative Property

The commutative property is a fundamental principle in mathematics. It states that the order of addition does not affect the sum. This principle applies broadly to real numbers and specifically to matrices. For instance, if you have two matrices, let's call them matrix A and matrix B, then the commutative property says that adding A to B gives the same result as adding B to A.

This concept is very helpful because it allows flexibility in mathematical operations. In everyday terms, it means it doesn't matter if you combine A first or B first - the outcome will remain consistent. The commutative property only applies to addition (and multiplication in some contexts) and not to subtraction or division. So when you're dealing with matrices, checking this property can confirm the consistency and correctness of your operations. For example, as demonstrated in the exercise, \( A + B = B + A \) highlights how matrices uphold this property.

Matrix Operations

Matrix operations include tasks such as addition, subtraction, and multiplication. Each operation has specific rules. When it comes to addition, two matrices can only be added if they have the same dimensions. This means that both matrices must have the same number of rows and columns.

When adding matrices, the operation is performed element-wise. This means that you add each element in one matrix to the corresponding element in the other. Mathematically, if you have two matrices, \( X \) and \( Y \), each of dimension \( m \times n \), their sum \( X+Y \) is given by \( Z_{ij} = X_{ij} + Y_{ij} \) for each corresponding element.

This process might look a bit complex at first, especially with larger matrices. However, matrix addition is simple once you become accustomed to it. It's like pairing socks - each sock (element) from one matrix pairs with its counterpart in the other matrix. Thus, access to matrix operations expands your ability to perform complex calculations across various scientific and engineering disciplines.

Matrices

Matrices are arrays of numbers organized in rows and columns. They are used extensively in various fields, including mathematics, physics, economics, and computer science. A single matrix could represent diverse things, like a system of equations, a transformation in space, or data arranged neatly for analysis.

Matrices come in different sizes, often referred to by their dimensions. For example, a 2x3 matrix has 2 rows and 3 columns. Understanding the size and configuration of matrices is crucial, especially when performing operations like addition and multiplication. Not all matrices can be added or multiplied; their compatibility depends on their dimensions.

Beyond operations, matrices have special types, such as square matrices, diagonal matrices, and identity matrices, each with unique properties and uses. Learning matrices opens up a world of systematic problem-solving, making complex systems easier to analyze and understand.