Problem 9 - 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 \(A+B+C+D=\mathbf{0}\).

Step-by-Step Solution

Answer

\(D = \begin{bmatrix} -3 & 1 & -9 \\ 0 & -4 & 1 \\ -3 & -1 & -5 \end{bmatrix}\).

Step 1: Understand the Goal

We need to find matrix \(D\) such that the sum of matrices \(A\), \(B\), \(C\), and \(D\) equals the zero matrix. Symbolically, this is given by the equation \(A + B + C + D = \mathbf{0}\), where \(\mathbf{0}\) is the zero matrix with the same dimensions as \(A\), \(B\), and \(C\).

Step 2: Define Zero Matrix

The zero matrix \(\mathbf{0}\) should have dimensions 3x3 since all matrices \(A\), \(B\), and \(C\) are 3x3. Thus, \(\mathbf{0} = \left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{array} \right]\).

Step 3: Add Matrices A, B, and C

To find \(D\), we first calculate \(A + B + C\). Calculate \(A + B\): \[A + B = \begin{bmatrix} 1 & 0 & 1 \ 2 & 3 & -1 \ 0 & -2 & 0 \end{bmatrix} + \begin{bmatrix} 1 & -1 & 4 \ -2 & 0 & -1 \ 1 & 3 & 3 \end{bmatrix} = \begin{bmatrix} 2 & -1 & 5 \ 0 & 3 & -2 \ 1 & 1 & 3 \end{bmatrix}\]Now add \(C\):\[A + B + C = \begin{bmatrix} 2 & -1 & 5 \ 0 & 3 & -2 \ 1 & 1 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 4 \ 0 & 1 & 1 \ 2 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix}\]

Step 4: Solve for D

Using the equation \(A + B + C + D = \mathbf{0}\), we solve for \(D\):\[D = \mathbf{0} - (A + B + C)\]Thus, \[D = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} - \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix} = \begin{bmatrix} -3 & 1 & -9 \ 0 & -4 & 1 \ -3 & -1 & -5 \end{bmatrix}\]

Step 5: Verify the Solution

Finally, verify that this calculation is correct by ensuring the matrix sum is zero:\[A + B + C + D = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix} + \begin{bmatrix} -3 & 1 & -9 \ 0 & -4 & 1 \ -3 & -1 & -5 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}\]This confirms that \(D \) is correct.

Key Concepts

Zero Matrix

A zero matrix is a special type of matrix where every single element is zero. This means, in any zero matrix, regardless of its size, all positions are filled with the number zero. Zero matrices play a crucial role in matrix algebra as they act as an additive identity for matrix addition. This means that when you add any matrix to a zero matrix of the same size, you get the original matrix back. For example, if you have a 3x3 zero matrix, it looks like this: \[\mathbf{0} = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}\]

• **Identity Element**: The zero matrix is like the number zero in regular arithmetic. Adding a zero matrix to any other matrix doesn't change that matrix.
• **Same Dimensions**: For two matrices to be added, both must have the same dimensions, which also applies to the zero matrix.

Understanding the concept of a zero matrix is essential because it allows simplification and balancing of matrix equations, just like in the problem where you solve for matrix \(D\) by ensuring the entire sum equals a zero matrix.

Matrix Addition

Matrix addition is the process of adding two matrices by adding the corresponding elements from each matrix. For the addition to be valid, both matrices must have the same dimensions, meaning they must have the same number of rows and columns. When adding two 3x3 matrices, each of the corresponding elements is summed up separately, resulting in another 3x3 matrix. The result when matrices \(A\), \(B\), and \(C\) from our exercise were added looks like this: \[\begin{bmatrix} 2 & -1 & 5 \ 0 & 3 & -2 \ 1 & 1 & 3 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 4 \ 0 & 1 & 1 \ 2 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 3 & -1 & 9 \ 0 & 4 & -1 \ 3 & 1 & 5 \end{bmatrix}\]

• **Simple Addition**: Just like adding numbers, align the matrices and add each pair of numbers in the same position.
• **Consistent Dimensions**: Remember, you can't add a 2x3 matrix to a 3x3 matrix.

This concept is fundamental in solving matrix equations where matrices need to be combined. Being familiar with matrix addition helps tremendously in understanding more complex operations and problem-solving.

3x3 Matrix

A 3x3 matrix is a specific type of matrix with three rows and three columns. Each position within the matrix can hold a numerical value, making a total of nine values in this kind of matrix. The format of a 3x3 matrix is structured like this:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]The 3x3 matrix form is extensively used in various mathematical and engineering disciplines.

• **Application**: The 3x3 matrix is crucial for transformations in geometry, computer graphics, and solving systems of linear equations.
• **Manipulation**: Operations such as addition, multiplication, or finding the determinant often involve such matrices.

Because 3x3 matrices are so common in problems, understanding their structure and how to manipulate them will help manage more complicated matrix calculations and applications across different fields.