Problem 7 - College Algebra

Question

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. $$ A=\left[\begin{array}{ll}{1} & {3} \\ {0} & {7}\end{array}\right], B=\left[\begin{array}{cc}{2} & {14} \\ {22} & {6}\end{array}\right], C=\left[\begin{array}{cc}{1} & {5} \\ {8} & {92} \\ {12} & {6}\end{array}\right], D=\left[\begin{array}{cc}{10} & {14} \\ {7} & {2} \\\ {5} & {61}\end{array}\right], E=\left[\begin{array}{cc}{6} & {12} \\ {14} & {5}\end{array}\right], F=\left[\begin{array}{cc}{0} & {9} \\ {78} & {17} \\\ {15} & {4}\end{array}\right] $$ $$ C+D $$

Step-by-Step Solution

Answer

Matrix addition possible. Result: \( \begin{bmatrix} 11 & 19 \\ 15 & 94 \\ 17 & 67 \end{bmatrix} \)

Step 1: Review Matrix Dimensions

Matrix addition is defined only for matrices with the same dimensions. First, check the dimensions of matrices \( C \) and \( D \). Matrix \( C \) is a \( 3 \times 2 \) matrix and matrix \( D \) is also a \( 3 \times 2 \) matrix.

Step 2: Confirm Addition Feasibility

Since both matrices \( C \) and \( D \) have the same dimensions (\( 3 \times 2 \)), the addition is defined and can be performed.

Step 3: Perform Element-wise Addition

Add the corresponding elements of matrices \( C \) and \( D \):\[\begin{bmatrix} 1 + 10 & 5 + 14 \ 8 + 7 & 92 + 2 \ 12 + 5 & 6 + 61\end{bmatrix} = \begin{bmatrix} 11 & 19 \ 15 & 94 \ 17 & 67\end{bmatrix}\]

Step 4: State the Result

The resulting matrix from adding matrices \( C \) and \( D \) is:\[\begin{bmatrix} 11 & 19 \ 15 & 94 \ 17 & 67\end{bmatrix}\]

Key Concepts

Matrix Dimensions

One of the fundamental aspects you need to understand when dealing with matrices is their dimensions. The dimension of a matrix is defined by the number of rows and columns it contains. For example, a matrix with 3 rows and 2 columns is referred to as a \(3 \times 2\) matrix. These dimensions play a crucial role, especially when performing operations like matrix addition.

For two matrices to be compatible for addition, they must have identical dimensions. If matrix \(C\) is a \(3 \times 2\) matrix, only matrices with the same \(3 \times 2\) layout can be added to it. This condition ensures that each element from one matrix can be paired with an element from the other matrix, keeping the operation logically consistent.

• Number of Rows: Both matrices must have the same number of rows.
• Number of Columns: Both matrices must have the same number of columns.

If the dimensions do not match, the operation is termed undefined, and the matrices cannot be added.

Element-wise Addition

Once you've confirmed that two matrices have the same dimensions, you can perform the addition through a method called element-wise addition. This method involves adding each corresponding element from the two matrices. It’s like adding numbers from two lists that are perfectly aligned in structure.

For example, if you have two matrices \(A\) and \(B\) with the same dimensions, the element at the first row and first column of matrix \(A\) will be added to the element at the first row and first column of matrix \(B\). This process continues for every position in the matrices.

• Step 1: Take the first element from the first matrix and the first element from the second matrix and add them together.
• Step 2: Move to the next element along the row and repeat the process.
• Step 3: Continue this process row by row until all elements are added.

By following these steps, the addition forms a new matrix where each entry is the sum of elements from the original matrices. This results in a new matrix with the same dimensions as the added matrices.

Matrix Operations

Matrix operations include a variety of calculations you can perform on matrices, such as addition, subtraction, multiplication, and more. Understanding these operations is key to using matrices effectively in mathematical contexts.

While matrix addition is one of the more straightforward operations—requiring simply that the matrices involved have the same dimensions—other operations like matrix multiplication have more complex rules due to the nature of how rows and columns interact. In that sense, addition is about direct correspondence between elements whereas multiplication involves more intricate coordination.

• Addition: Can be performed if matrices have the same dimensions; results in a matrix of the same size.
• Subtraction: Similar to addition; requires matrices of the same dimensions.
• Multiplication: More complex; not dimension-dependent in the same way as addition. Requires that the number of columns in the first matrix equals the number of rows in the second.

Understanding these operations expands the range of problems you can solve with matrices, allowing you to handle everything from simple addition to complex linear algebra problems.