Problem 6 - Calculus for Biology and Medicine

Question

Let $$A=\left[\begin{array}{rr}-1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right], \quad C=\left[\begin{array}{rr}1 & -2 \\ -1 & -1\end{array}\right]$$ Show that \(2(A+B)=2 A+2 B\).

Step-by-Step Solution

Answer

Yes, the equation \(2(A + B) = 2A + 2B\) holds true.

Step 1: Add Matrices A and B

First, we need to add matrices \(A\) and \(B\). The matrices are:\[A = \begin{bmatrix} -1 & 2 \ 0 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \ 2 & 0 \end{bmatrix}\]To add matrices \(A\) and \(B\), add corresponding elements:\[A + B = \begin{bmatrix} -1 + 0 & 2 + 1 \ 0 + 2 & -3 + 0 \end{bmatrix} = \begin{bmatrix} -1 & 3 \ 2 & -3 \end{bmatrix}\]

Step 2: Multiply the Result by 2

Now, multiply the result from Step 1 by 2:\[2(A + B) = 2 \times \begin{bmatrix} -1 & 3 \ 2 & -3 \end{bmatrix} = \begin{bmatrix} 2(-1) & 2(3) \ 2(2) & 2(-3) \end{bmatrix} = \begin{bmatrix} -2 & 6 \ 4 & -6 \end{bmatrix}\]

Step 3: Compute 2A

First, compute \(2A\). Given matrix \(A\):\[A = \begin{bmatrix} -1 & 2 \ 0 & -3 \end{bmatrix}\]Multiply each element by 2:\[2A = 2 \times \begin{bmatrix} -1 & 2 \ 0 & -3 \end{bmatrix} = \begin{bmatrix} -2 & 4 \ 0 & -6 \end{bmatrix}\]

Step 4: Compute 2B

Next, compute \(2B\). Given matrix \(B\):\[B = \begin{bmatrix} 0 & 1 \ 2 & 0 \end{bmatrix}\]Multiply each element by 2:\[2B = 2 \times \begin{bmatrix} 0 & 1 \ 2 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 2 \ 4 & 0 \end{bmatrix}\]

Step 5: Add 2A and 2B

Finally, we add the results of \(2A\) and \(2B\):\[2A + 2B = \begin{bmatrix} -2 & 4 \ 0 & -6 \end{bmatrix} + \begin{bmatrix} 0 & 2 \ 4 & 0 \end{bmatrix} = \begin{bmatrix} -2+0 & 4+2 \ 0+4 & -6+0 \end{bmatrix} = \begin{bmatrix} -2 & 6 \ 4 & -6 \end{bmatrix}\]

Step 6: Conclusion: Verify the Equation

Both computations give us the same result:\[2(A + B) = \begin{bmatrix} -2 & 6 \ 4 & -6 \end{bmatrix}, \quad 2A + 2B = \begin{bmatrix} -2 & 6 \ 4 & -6 \end{bmatrix}\]This confirms that:\[2(A + B) = 2A + 2B\]

Key Concepts

Matrix Addition

In the world of matrix algebra, matrix addition is one of the simplest operations. It involves adding two matrices element by element.
To add two matrices, make sure they have the same dimensions. This means the number of rows and columns in both matrices must be equal.
If you have two matrices, like matrix \( A \) and matrix \( B \), their addition is expressed as \( A + B \). You perform addition by summing the corresponding elements from each matrix:

• The element in the first row and first column of \( A \) is added to the element in the first row and first column of \( B \).
• This pattern is continued for each element in the matrix, ensuring no matching element is left behind.
  

In our example, we added matrices \( A \) and \( B \) to get a new matrix with each entry being the sum of corresponding entries from \( A \) and \( B \). This fundamental operation lays the groundwork for more advanced concepts in matrix algebra.

Matrix Multiplication

Matrix multiplication is a slightly more complex operation than addition. It involves the multiplication of a matrix by a scalar, a single number, or by another matrix.
When multiplying a matrix by a scalar, you simply multiply each element of the matrix by that scalar.
For example, if you multiply matrix \( A \) by \( 2 \), each element of \( A \) is multiplied by \( 2 \). This is what you do when you see expressions like \( 2A \) or \( 2B \).
Another way is to multiply two matrices, which requires careful attention to their dimensions. The number of columns in the first matrix must match the number of rows in the second matrix. The resultant matrix will have the same number of rows as the first matrix and columns as the second one:

• Each element of the resultant matrix is calculated by taking the dot product of the corresponding row from the first matrix and column from the second matrix. This involves multiplying corresponding elements and then summing them up.
• This process continues for each row-column pair in the matrices.
  

In our exercise, we focused on scalar multiplication as part of proving the equation \( 2(A+B) = 2A + 2B \). This involved scalar multiplication of matrices \( A \) and \( B \) and the result of their addition, highlighting the distributive property over matrix addition.

Properties of Matrices

Matrices have several important properties that aid in their manipulation and analysis. These properties make matrices essential tools in various fields, including mathematics, engineering, and computer science.
Some of the key properties include:

• Commutative Property in Addition: This states that for any two matrices \( A \) and \( B \) of the same dimensions, \( A + B = B + A \).
• Associative Property in Addition: This property indicates that \( (A + B) + C = A + (B + C) \), emphasizing the order in which matrices are added doesn’t change the result.
• Distributive Property: The distributive property plays a crucial role in our original exercise. It states that \( c(A + B) = cA + cB \) for a scalar \( c \). Essentially, multiplying a sum of matrices by a scalar is the same as multiplying each matrix by the scalar first and then adding them.
  

Understanding these properties is essential for efficiently handling complex matrix equations and performing operations correctly. In the exercise, using the distributive property confirmed that \( 2(A+B) = 2A + 2B \), demonstrating the beauty and utility of these mathematical constructs.