Problem 8 - Calculus for Biology and Medicine

Question

$$ \begin{array}{r} A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ 2 & 4 & 1 \\ 0 & -2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 5 & -1 & 4 \\ 2 & 0 & 1 \\ 1 & -3 & -3 \end{array}\right], \\ \quad C=\left[\begin{array}{rrr} -2 & 0 & 4 \\ 1 & -3 & 1 \\ 0 & 0 & 2 \end{array}\right] \end{array} $$ $$ \text { Find } 3 C-B+\frac{1}{2} A $$

Step-by-Step Solution

Answer

The result is \( \begin{bmatrix} -10.5 & 2.5 & 7.5 \\ 2 & -7 & 2.5 \\ -1 & 2 & 10 \end{bmatrix} \).

Step 1: Multiply Matrix C by 3

To begin, multiply each element of matrix \( C \) by 3. Matrix \( C \) is: \[ C = \begin{bmatrix} -2 & 0 & 4 \ 1 & -3 & 1 \ 0 & 0 & 2 \end{bmatrix} \]Multiplying \( C \) by 3 gives us:\[ 3C = \begin{bmatrix} -6 & 0 & 12 \ 3 & -9 & 3 \ 0 & 0 & 6 \end{bmatrix} \]

Step 2: Divide Matrix A by 2

The next step is to divide each element of matrix \( A \) by 2.Matrix \( A \) is: \[ A = \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} \]Dividing \( A \) by 2 gives us:\[ \frac{1}{2}A = \begin{bmatrix} 0.5 & 1.5 & -0.5 \ 1 & 2 & 0.5 \ 0 & -1 & 1 \end{bmatrix} \]

Step 3: Subtract Matrix B from 3C

Now, subtract matrix \( B \) from \( 3C \).Matrix \( B \) is: \[ B = \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} \]Perform the subtraction:\[ 3C - B = \begin{bmatrix} -6-5 & 0+1 & 12-4 \ 3-2 & -9-0 & 3-1 \ 0-1 & 0+3 & 6+3 \end{bmatrix} = \begin{bmatrix} -11 & 1 & 8 \ 1 & -9 & 2 \ -1 & 3 & 9 \end{bmatrix} \]

Step 4: Add \( \frac{1}{2}A \) to \( 3C-B \)

Finally, add the result from Step 2 to the result from Step 3.The matrices are:\[ 3C - B = \begin{bmatrix} -11 & 1 & 8 \ 1 & -9 & 2 \ -1 & 3 & 9 \end{bmatrix} \] and \[ \frac{1}{2}A = \begin{bmatrix} 0.5 & 1.5 & -0.5 \ 1 & 2 & 0.5 \ 0 & -1 & 1 \end{bmatrix} \]Adding them together gives:\[ (3C - B) + \frac{1}{2}A = \begin{bmatrix} -11+0.5 & 1+1.5 & 8-0.5 \ 1+1 & -9+2 & 2+0.5 \ -1+0 & 3-1 & 9+1 \end{bmatrix} = \begin{bmatrix} -10.5 & 2.5 & 7.5 \ 2 & -7 & 2.5 \ -1 & 2 & 10 \end{bmatrix} \]

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation where two matrices are combined to produce a new matrix. The process involves taking the rows of the first matrix and "dot multiplying" each by the columns of the second matrix. This means you take each element of the row, multiply by the corresponding element in the column, and sum the products. The result fills the corresponding entry in the new matrix. This can only be done if the number of columns in the first matrix equals the number of rows in the second matrix. For example, if you have a 3x2 matrix (3 rows, 2 columns) and you want to multiply it with a 2x3 matrix, you can, because the inner dimensions (the 2s) match.

The product will be a 3x3 matrix because the outer dimensions (the 3s) determine the size of the product matrix. Important points to remember:

• Matrix multiplication is not commutative: \( AB eq BA \).
• It is associative, meaning: \( A(BC) = (AB)C \).
• The identity matrix acts as the multiplicative identity, so for any matrix \( A \), \( AI = IA = A \), where \( I \) is the identity matrix of compatible dimensions.

Understanding these properties will help you solve a wide range of problems involving matrix multiplication.

Matrix Addition

Matrix addition is a straightforward operation where you add corresponding elements of two matrices to obtain a new matrix. This means you take each element from one matrix and add it to the respective element in the second matrix. The result of each addition operation becomes an element of a new matrix. It is important to note that for this operation to work, both matrices must have the same dimensions. For example, a 3x3 matrix can only be added to another 3x3 matrix.

In calculations, matrix addition follows some fundamental rules:

• It is commutative: \( A + B = B + A \).
• It is associative: \( (A + B) + C = A + (B + C) \).
• The zero matrix, where each element is zero, acts as the additive identity: \( A + 0 = A \).

Matrix addition is commonly used in various fields like computer graphics and solving systems of linear equations.

Matrix Subtraction

Matrix subtraction operates similarly to matrix addition but instead involves subtracting corresponding elements. Just as with addition, the two matrices you wish to subtract must have the same dimensions. For instance, every element in the first matrix is subtracted from the corresponding element in the second matrix, and the results form a new matrix that represents the difference.

Key points to understand about matrix subtraction:

• Matrix subtraction is NOT commutative: \( A - B eq B - A \).
• It can still be associative, provided you are consistent with the order: \( (A - B) - C = A - (B + C) \).
• The zero matrix serves as a neutral point:\( A - 0 = A \).

Matrix subtraction is often utilized in solving problems related to differential equations, among other mathematical areas.