Problem 10 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

Matrix \(D\) is \(\begin{bmatrix} 1 & -2 & 7 \\ -6 & -3 & -1 \\ 2 & 8 & 6 \end{bmatrix}\).

Step 1: Write the Given Equation

We need to determine the matrix \(D\) for the equation: \[ A + 4B = 2(A + B) + D \]

Step 2: Simplify the Right Side of the Equation

Firstly, simplify the expression on the right side of the equation:- Distribute the 2 into \((A + B)\): - \(2(A + B) = 2A + 2B\)Thus, the equation becomes:\[ A + 4B = 2A + 2B + D \]

Step 3: Isolate Matrix D

Now, to isolate \(D\), move the terms \(2A\) and \(2B\) to the left side of the equation:\[ A + 4B - 2A - 2B = D \]Combine like terms:\[ (A - 2A) + (4B - 2B) = D \]

Step 4: Simplify Left Side

Calculate the differences:- \(A - 2A = -A\)- \(4B - 2B = 2B\)Thus:\[ -A + 2B = D \]

Step 5: Substitute Matrix Values and Calculate

Substitute the matrices for \(A\) and \(B\) and perform the operations:- Let \(A = \begin{bmatrix} 1 & 0 & 1 \ 2 & 3 & -1 \ 0 & -2 & 0 \end{bmatrix}\)- Let \(B = \begin{bmatrix} 1 & -1 & 4 \ -2 & 0 & -1 \ 1 & 3 & 3 \end{bmatrix}\)Compute \(-A\):\[-A = \begin{bmatrix} -1 & 0 & -1 \ -2 & -3 & 1 \ 0 & 2 & 0 \end{bmatrix}\]Compute \(2B\):\[2B = \begin{bmatrix} 2 \times 1 & 2 \times -1 & 2 \times 4 \ 2 \times -2 & 2 \times 0 & 2 \times -1 \ 2 \times 1 & 2 \times 3 & 2 \times 3 \end{bmatrix} = \begin{bmatrix} 2 & -2 & 8 \ -4 & 0 & -2 \ 2 & 6 & 6 \end{bmatrix}\]Finally, calculate \(D\):\[D = -A + 2B = \begin{bmatrix} -1 & 0 & -1 \ -2 & -3 & 1 \ 0 & 2 & 0 \end{bmatrix} + \begin{bmatrix} 2 & -2 & 8 \ -4 & 0 & -2 \ 2 & 6 & 6 \end{bmatrix} = \begin{bmatrix} 1 & -2 & 7 \ -6 & -3 & -1 \ 2 & 8 & 6 \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations serve as the fundamental actions we perform on matrices. These operations include addition, subtraction, and multiplication of matrices, as well as scalar multiplication. They extend the basic arithmetic operations to a more structured form, allowing us to work with complex data sets in a systematic way.

When adding or subtracting matrices, they must have the same dimensions. For example, two matrices of size 3x3 can be added if they have the same number of rows and columns. The corresponding elements of the matrices are added together. In the equation given in the exercise, notice how matrices \(A\) and \(B\) are added directly:

• For addition: \(A + B = \begin{bmatrix} 1+1 & 0+(-1) & 1+4 \ 2+(-2) & 3+0 & -1+(-1) \ 0+1 & -2+3 & 0+3 \end{bmatrix}\)
• For subtraction, each element of the first matrix is subtracted by the corresponding element of the second matrix.

Just as in regular arithmetic, each of these operations follows specific rules and conventions which help streamline complex calculations. Understanding these matrix operations is crucial for progressing in matrix algebra.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (or linear spaces), and linear transformations, which include matrices. It forms the backbone of many applications in various fields such as computer science, engineering, and physics due to its powerful methods for solving systems of linear equations.

Matrices are key components in linear algebra as they represent linear transformations and systems of linear equations. They allow for compact storage and manipulation of data. For example, matrix multiplication can be used to apply multiple linear transformations sequentially to a dataset.

The fundamental concepts of linear algebra, like linear combinations, linear independence, and basis, revolve around the study and understanding of vector spaces. Matrices make it possible to transition between different bases, perform transformations, and solve equations efficiently.

• Solutions to systems of linear equations: Represented by augmenting matrices and augmented to nascent forms like reduced row echelon format for solving equations.
• Applications in eigenvalues and eigenvectors: Critical for understanding systems dynamics and stability.

Linear algebra helps bridge the gap between basic algebraic principles and more complex mathematics, enabling powerful problem-solving capabilities.

Matrix Equations

Matrix equations are systems structured similarly to conventional equations but involve matrices as the unknown components to be solved. Similar to algebraic equations, matrix equations involve finding one or more unknown matrices that satisfy a given condition.

In the exercise, the given equation \(A + 4B = 2(A + B) + D\) is a matrix equation. The goal here is to solve for the matrix \(D\). To isolate \(D\), we rearrange terms and use matrix operations like distribution and subtraction.

Let's explore how we handle these equations:

• Distribution: Similar to algebraic expressions, distribute constants across matrix terms. For example, \(2(A + B) = 2A + 2B\).
• Solving: Rearrange terms to solve for the unknown matrix, just like solving for \(x\) in an algebraic equation. Move known matrices to one side of the equation to isolate the unknown matrix.
• Substitution: Substitute numerical values for matrix components, allowing calculation of the final result.

Mastering the manipulation and solving of matrix equations is vital because they appear frequently in fields such as physics, economics, and statistics, offering a way to handle large systems in an organized manner.