Problem 12 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\((A+B)+C = A+(B+C)\) holds as both result in the same matrix.

Step 1: Compute A + B

Add the corresponding elements of matrices A and B. \[A+B = \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} + \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} = \begin{bmatrix} 1+5 & 3-1 & -1+4 \ 2+2 & 4+0 & 1+1 \ 0+1 & -2-3 & 2-3 \end{bmatrix} = \begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix} \]

Step 2: Compute (A + B) + C

Add the matrix C to the result of (A + B) found in Step 1.\[(A+B) + C = \begin{bmatrix} 6 & 2 & 3 \ 4 & 4 & 2 \ 1 & -5 & -1 \end{bmatrix} + \begin{bmatrix} -2 & 0 & 4 \ 1 & -3 & 1 \ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 6-2 & 2+0 & 3+4 \ 4+1 & 4-3 & 2+1 \ 1+0 & -5+0 & -1+2 \end{bmatrix} = \begin{bmatrix} 4 & 2 & 7 \ 5 & 1 & 3 \ 1 & -5 & 1 \end{bmatrix} \]

Step 3: Compute B + C

Add the corresponding elements of matrices B and C.\[B+C = \begin{bmatrix} 5 & -1 & 4 \ 2 & 0 & 1 \ 1 & -3 & -3 \end{bmatrix} + \begin{bmatrix} -2 & 0 & 4 \ 1 & -3 & 1 \ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 5-2 & -1+0 & 4+4 \ 2+1 & 0-3 & 1+1 \ 1+0 & -3+0 & -3+2 \end{bmatrix} = \begin{bmatrix} 3 & -1 & 8 \ 3 & -3 & 2 \ 1 & -3 & -1 \end{bmatrix} \]

Step 4: Compute A + (B + C)

Add matrix A to the result of (B + C) found in Step 3.\[A + (B+C) = \begin{bmatrix} 1 & 3 & -1 \ 2 & 4 & 1 \ 0 & -2 & 2 \end{bmatrix} + \begin{bmatrix} 3 & -1 & 8 \ 3 & -3 & 2 \ 1 & -3 & -1 \end{bmatrix} = \begin{bmatrix} 1+3 & 3-1 & -1+8 \ 2+3 & 4-3 & 1+2 \ 0+1 & -2-3 & 2-1 \end{bmatrix} = \begin{bmatrix} 4 & 2 & 7 \ 5 & 1 & 3 \ 1 & -5 & 1 \end{bmatrix} \]

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

See that both steps 2 and 4 result in the same matrix:\[\begin{bmatrix} 4 & 2 & 7 \ 5 & 1 & 3 \ 1 & -5 & 1 \end{bmatrix} \]This confirms that the associativity property holds, thus \((A+B)+C = A+(B+C)\).

Key Concepts

Associative Property

The associative property is a fundamental concept in mathematics that applies to various operations, including matrix addition. This property dictates that the manner in which numbers or elements are grouped in an operation does not affect the final result. In simpler terms, if you change the grouping of the numbers being added, the sum remains the same. For three matrices A, B, and C, the associative property for addition can be described as:

• (A + B) + C = A + (B + C)

This means that we can first add A and B, and subsequently add C to the result, or we can first add B and C, and then add A to this new sum. Regardless of which order we perform the operations, the total sum will be identical.
In the exercise, this property is used to show that the sum of the matrices remains consistent no matter how the matrices are associated or grouped during the addition.

Matrix Addition

Matrix addition is a straightforward process where we add corresponding elements of the matrices. For two matrices to be compatible for addition, they must have the same dimensions, meaning they have the same number of rows and columns. The result of this addition is a new matrix, where each element is the sum of the corresponding elements from the original matrices.

Let's recall the matrices given in the exercise:

• Matrix A: a 3x3 matrix
• Matrix B: another 3x3 matrix
• Matrix C: yet another 3x3 matrix

Since all matrices involved in this exercise are of the same size, they can be added directly by summing the respective elements. If we visualize matrix addition, imagine lining up the matrices and adding each position, like matching socks and putting them together to find a complete pair. This simple operation builds a foundation for more advanced matrix operations.

Step-by-Step Solution

To thoroughly understand the solution, breaking it down step-by-step eases comprehension. The exercise involves proving the associative property with three matrices A, B, and C:

• First Step: Calculate A + B by adding each corresponding element of A and B.
• Second Step: Then, compute (A + B) + C by adding the resulting matrix from step one with matrix C.
• Third Step: Calculate B + C by adding each corresponding element of B and C separately.
• Fourth Step: Finally, compute A + (B + C) by adding matrix A to the result of B + C achieved in the previous step.
• Fifth Step: Compare the results of (A + B) + C and A + (B + C). If they match, it confirms the associative property for matrix addition.

By following these steps, we can visualize the process. This structured approach demystifies the concept, making what is sometimes an abstract operation seem more tangible and manageable. This method not only proves the property but also helps enhance understanding of matrices behavior in algebraic operations.