Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$\left[\begin{array}{rrrr}9 & 4 & 1 & -2 \\ 5 & -6 & 3 & 4 \\ 2 & -5 & 1 & 2\end{array}\right]-\left[\begin{array}{rrrr}-2 & 5 & 1 & 3 \\ 0 & 1 & 0 & 2 \\\ -8 & 3 & 2 & 1\end{array}\right]+\left[\begin{array}{rrrr}2 & 4 & 0 & 3 \\\ 4 & -5 & 1 & 6 \\ 2 & -3 & 0 & 8\end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is: \[ \begin{bmatrix} 13 & 3 & 0 & -2 \\ 9 & -12 & 4 & 8 \\ 12 & -11 & -1 & 9 \end{bmatrix} \]

Step 1: Check Matrix Dimensions

First, ensure the matrices are compatible for addition and subtraction. Each matrix is a 3x4 matrix, which means we can perform these operations element-wise.

Step 2: Subtract First and Second Matrices

Subtract corresponding elements of the second matrix from the first matrix: \[ \begin{bmatrix} 9 - (-2) & 4 - 5 & 1 - 1 & -2 - 3 \ 5 - 0 & -6 - 1 & 3 - 0 & 4 - 2 \ 2 - (-8) & -5 - 3 & 1 - 2 & 2 - 1 \end{bmatrix} \]Calculate:\[ \begin{bmatrix} 11 & -1 & 0 & -5 \ 5 & -7 & 3 & 2 \ 10 & -8 & -1 & 1 \end{bmatrix} \]

Step 3: Add Result from Step 2 with Third Matrix

Add each element from the result of Step 2 with the corresponding element of the third matrix:\[ \begin{bmatrix} 11 + 2 & -1 + 4 & 0 + 0 & -5 + 3 \ 5 + 4 & -7 - 5 & 3 + 1 & 2 + 6 \ 10 + 2 & -8 - 3 & -1 + 0 & 1 + 8 \end{bmatrix} \]Calculate:\[ \begin{bmatrix} 13 & 3 & 0 & -2 \ 9 & -12 & 4 & 8 \ 12 & -11 & -1 & 9 \end{bmatrix} \]

Step 4: Conclusion

After performing all operations, the resulting matrix is: \[ \begin{bmatrix} 13 & 3 & 0 & -2 \ 9 & -12 & 4 & 8 \ 12 & -11 & -1 & 9 \end{bmatrix} \]

Key Concepts

Matrix Addition

Matrix addition is a fundamental concept in linear algebra, where two matrices of the same dimensions can be added together to produce a new matrix. This operation is done by adding corresponding elements of each matrix. For example, consider two matrices, A and B, both of size 3x4. The resulting matrix C from their addition will also be a 3x4 matrix.

When performing matrix addition, there are a few things to keep in mind:

• The matrices must have the same dimensions; otherwise, addition is not possible.
• Elements are added by their position, meaning the element in the first row and first column of each matrix is added together, and so forth for each element.

Matrix addition is commutative, which means that the order in which two matrices are added does not change the result. For instance, A + B is the same as B + A.

Upon adding multiple matrices, like in this exercise, each pair of corresponding elements from the intermediate results is summed with those of another matrix until a final matrix is obtained.

Matrix Subtraction

Matrix subtraction, just like matrix addition, requires that the matrices involved are of the same size. This ensures that corresponding elements can be appropriately subtracted from one another. In our case, subtracting matrix B from matrix A results in a new 3x4 matrix C, where each element in C is the result of the corresponding element in A minus the element in B.

The key points to remember about matrix subtraction are:

• Both matrices must have identical dimensions (same number of rows and columns).
• Subtract elements located in the same positions of the matrices, such as the element in the first row and first column of A with that of B.

Matrix subtraction is not commutative. This means that the result of A - B is not the same as B - A. This difference highlights the importance of the order of the matrices during subtraction.

In an exercise where multiple matrices are involved, each subtraction is intermediate to another operation, often followed (or preceded) by addition, as seen in the steps.

Element-wise Operations

Element-wise operations in matrices refer to performing arithmetic actions, such as addition or subtraction, solely on corresponding elements of matrices. This approach assumes that the matrices are of the same dimension, a prerequisite for conducting element-wise operations.

For any given operation, the formula typically involves taking an element from one position in the first matrix and performing the operation with the element in the same position of the other matrix. This results in a new matrix that reflects these individual operations.

• For example, if matrices A and B both have a 3x4 dimension, then element-wise addition or subtraction will involve each of the elements in one position being operated on with the corresponding element in the other matrix.
• This type of operation is straightforward but essential for coordinating multiple matrices in larger computations or systems.

Element-wise operations ensure flexibility and precision when working across datasets captured in matrices, allowing for direct application of arithmetic operations per individual matrix entries.

Throughout the exercise, element-wise subtraction and addition were utilized to achieve the final resultant matrix proficiently, adhering to the foundational rule of equivalency in matrix dimensions.