Problem 10 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

Matrix \( D = \begin{bmatrix} 9 & -5 & 9 \\ 2 & -4 & 1 \\ 2 & -4 & -8 \end{bmatrix} \).

Step 1: Simplify the Left Side of the Equation

Compute \( A + 4B \). Start by multiplying matrix \( B \) by 4 and then adding the result to matrix \( A \).First, calculate \( 4B \):\[4B = 4 \times \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} = \begin{bmatrix} 20 & -4 & 16 \ 8 & 0 & 4 \ 4 & -12 & -12 \end{bmatrix}\]Now, calculate \( A + 4B \):\[A + 4B = \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} + \begin{bmatrix} 20 & -4 & 16 \ 8 & 0 & 4 \ 4 & -12 & -12 \end{bmatrix} = \begin{bmatrix} 21 & -1 & 15 \ 10 & 4 & 5 \ 4 & -14 & -10 \end{bmatrix}\]

Step 2: Simplify the Right Side of the Equation

Compute \( 2(A + B) \) and simplify \( 2(A + B) + D \).First, calculate \( A + B \):\[A + B = \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} + \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} = \begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix}\]Now, compute \( 2(A + B) \):\[2(A + B) = 2 \times \begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix} = \begin{bmatrix} 12 & 4 & 6 \ 8 & 8 & 4 \ 2 & -10 & -2 \end{bmatrix}\]

Step 3: Solve for Matrix D

Use the equation \( A + 4B = 2(A + B) + D \) to solve for \( D \).Subtract \( 2(A + B) \) from both sides to isolate \( D \):\[D = (A + 4B) - 2(A + B)\]Substitute the matrices from the previous steps:\[D = \begin{bmatrix} 21 & -1 & 15 \ 10 & 4 & 5 \ 4 & -14 & -10 \end{bmatrix} - \begin{bmatrix} 12 & 4 & 6 \ 8 & 8 & 4 \ 2 & -10 & -2 \end{bmatrix} = \begin{bmatrix} 9 & -5 & 9 \ 2 & -4 & 1 \ 2 & -4 & -8 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra that involves the element-wise summation of two matrices with the same dimensions. When adding two matrices, each element in one matrix is added to the corresponding element in the second matrix. For instance, if you have matrices \( A \) and \( B \) both of size \( m \times n \), their sum \( A + B \) will also result in an \( m \times n \) matrix.

To perform matrix addition:

• Ensure both matrices are of the same dimension. Adding matrices of different dimensions is undefined.
• Add each corresponding element from both matrices together.

For example, let's say \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \). The result of \( A + B \) would be \( \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix} \).
Matrix addition is commutative, meaning \( A + B = B + A \), and associative, meaning \( (A + B) + C = A + (B + C) \). These properties make it a flexible tool in mathematical computations.

Matrix Multiplication

Matrix multiplication is a more complex operation than matrix addition. It involves taking the dot product of the rows of the first matrix with the columns of the second matrix. This operation requires the number of columns in the first matrix to be equal to the number of rows in the second matrix. The resulting product will be a new matrix whose dimensions are defined by the number of rows of the first matrix and the number of columns of the second matrix.

Here's how matrix multiplication is performed:

• Take each row in the first matrix and each column in the second matrix.
• Multiply elements pairwise from the row and column, then sum up these products.

For instance, if you multiply matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) by matrix \( B = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix} \), the resulting product \( AB \) will be \( \begin{bmatrix} (1 \cdot 2) + (2 \cdot 1) & (1 \cdot 0) + (2 \cdot 3) \ (3 \cdot 2) + (4 \cdot 1) & (3 \cdot 0) + (4 \cdot 3) \end{bmatrix} = \begin{bmatrix} 4 & 6 \ 10 & 12 \end{bmatrix} \).
It's important to note that, unlike addition, matrix multiplication is not commutative (i.e., \( AB eq BA \) in general). However, it does follow the associative and distributive laws.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vector spaces and the linear mappings between them. It involves vectors, matrices, systems of linear equations, and forms the foundation for much of modern mathematical, scientific, and engineering analyses.

Key aspects of linear algebra include:

• Vector Spaces: Collections of vectors where vector addition and scalar multiplication are defined.
• Matrices: Rectangular arrays of numbers representing linear transformations or systems of equations.
• Linear Transformations: Functions that map vectors to other vectors, preserving vector addition and scalar multiplication.

Linear algebra is central to many areas of mathematics and its applications. For instance, solutions to systems of linear equations are often found using matrices. Linear transformations are used to rotate, scale, and transform data in spaces.
Linear algebra provides the tools to work with large sets of equations compactly and efficiently. An understanding of linear algebra concepts is crucial in fields that involve computations and data processing, such as computer graphics, machine learning, and quantum physics. With its vast application scope, mastering linear algebra is a valuable endeavor in a wide range of academic and professional fields.