Problem 12 - 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 \((A+B)+C=A+(B+C)\).

Step-by-Step Solution

Answer

\((A + B) + C = A + (B + C)\) is true; matrix addition is associative.

Step 1: Compute A + B

First, we add matrices \( A \) and \( B \). To do this, add the corresponding elements from each matrix:\[A = \begin{bmatrix} 1 & 0 & 1 \ 2 & 3 & -1 \ 0 & -2 & 0 \end{bmatrix},\quadB = \begin{bmatrix} 1 & -1 & 4 \ -2 & 0 & -1 \ 1 & 3 & 3 \end{bmatrix}\]Perform element-wise addition:\[A + B = \begin{bmatrix} 1+1 & 0+(-1) & 1+4 \ 2+(-2) & 3+0 & -1+(-1) \ 0+1 & -2+3 & 0+3 \end{bmatrix} = \begin{bmatrix} 2 & -1 & 5 \ 0 & 3 & -2 \ 1 & 1 & 3 \end{bmatrix}\]

Step 2: Compute (A + B) + C

Using the result from Step 1, add the matrix \( C \) to \( A + B \):\[C = \begin{bmatrix} 1 & 0 & 4 \ 0 & 1 & 1 \ 2 & 0 & 2 \end{bmatrix}\]Perform element-wise addition with \( A + B \):\[(A + B) + C = \begin{bmatrix} 2+1 & -1+0 & 5+4 \ 0+0 & 3+1 & -2+1 \ 1+2 & 1+0 & 3+2 \end{bmatrix} = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix}\]

Step 3: Compute B + C

Next, add matrices \( B \) and \( C \). Perform element-wise addition:\[B + C = \begin{bmatrix} 1+1 & -1+0 & 4+4 \ -2+0 & 0+1 & -1+1 \ 1+2 & 3+0 & 3+2 \end{bmatrix} = \begin{bmatrix} 2 & -1 & 8 \ -2 & 1 & 0 \ 3 & 3 & 5 \end{bmatrix}\]

Step 4: Compute A + (B + C)

Using the result from Step 3, add the matrix \( A \) to \( B + C \):\[A = \begin{bmatrix} 1 & 0 & 1 \ 2 & 3 & -1 \ 0 & -2 & 0 \end{bmatrix}\]Perform element-wise addition:\[A + (B + C) = \begin{bmatrix} 1+2 & 0+(-1) & 1+8 \ 2+(-2) & 3+1 & -1+0 \ 0+3 & -2+3 & 0+5 \end{bmatrix} = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix}\]

Step 5: Compare (A + B) + C and A + (B + C)

After calculation, we see:- \((A + B) + C = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix}\)- \(A + (B + C) = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix}\)Since both results are identical, it confirms that matrix addition is associative.

Key Concepts

Associative Property

In matrix addition, the associative property is a fundamental concept that reveals the flexibility of adding matrices. It essentially states that the order in which matrices are added does not affect the final result. This property helps simplify complex computations by allowing us to group matrices differently. Consider three matrices: \(A\), \(B\), and \(C\). According to the associative property of addition, the equation \((A + B) + C = A + (B + C)\) always holds true. This means that if you first add matrices \(A\) and \(B\), and then add their sum to matrix \(C\), it will produce the same result as first adding matrices \(B\) and \(C\), and then adding matrix \(A\) to their sum.

Understanding this property is crucial because it ensures that our calculations remain consistent, no matter how we group the matrices during addition. This is especially useful in larger computations, where simplifying the process can save you time and help avoid mistakes.

Some key points about the associative property in matrix addition include:

• It applies to any number of matrices as long as they have the same dimensions.
• The property provides flexibility and efficiency in mathematical computations.
• It is crucial for theoretical proofs and practical problem-solving in linear algebra.

Element-Wise Addition

Element-wise addition is the method used to add two matrices. It's quite simple yet vital to understand how matrix addition works. In this context, each element of one matrix is added to the corresponding element of another matrix. To give an example, if we have matrices \(A\) and \(B\), the element in the first row and first column of \(A\) is added to the element in the same position of \(B\). This process is repeated for all elements in the matrices.

Here's how you can think of it:

• Step-by-step approach: For each pair of corresponding elements (i.e., those in the same position), you simply add them together.
• Matrix dimensions: For element-wise addition to be possible, both matrices must have the same dimensions (both need to have the same number of rows and columns).
• Resultant matrix: The result will be another matrix, often called the sum matrix, which contains the sums of the corresponding elements from the original matrices.

Using element-wise addition is a straightforward and reliable method, ensuring that every paired element is treated consistently and systematically. It is the backbone of many matrix operations in linear algebra.

Matrices

Matrices are rectangular arrays of numbers organized in rows and columns and are fundamental objects in linear algebra. They serve as a powerful tool to represent and solve systems of linear equations, transformations, and more, in various fields such as engineering, computer science, and physics.

The dimensions of a matrix are represented as 'rows \(\times\) columns'. For example, a matrix with 3 rows and 3 columns is known as a 3x3 matrix. Each entry in a matrix is called an element, and these elements can be added, subtracted, or multiplied according to matrix algebra rules.

Here's what you need to know about matrices:

• Types of matrices: There are several types of matrices, including square matrices, identity matrices, zero matrices, and diagonal matrices, each with unique properties.
• Applications: Matrices are used to describe systems of linear equations, perform transformations in graphics, among others.
• Operations: Besides matrix addition, matrices can be multiplied, transposed, and inverted, each operation having specific rules and applications.

Understanding matrices is crucial for anyone diving into higher mathematics or fields that use computational mathematics because they provide a structured and efficient approach to handling large sets of equations and computations.