Problem 10 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The result of adding matrices C and F is \(\begin{bmatrix} 1 & 14 \\ 86 & 109 \\ 27 & 10 \end{bmatrix}\).

Step 1: Determine Matrix Dimensions

First, identify the dimensions of the matrices involved. Matrix \(C\) has dimensions \(3 \times 2\) (3 rows and 2 columns) and matrix \(F\) has dimensions \(3 \times 2\).

Step 2: Check Compatibility for Addition

Matrix addition is defined only when both matrices have the same dimensions. In this case, both \(C\) and \(F\) have the dimensions \(3 \times 2\), so the addition is defined.

Step 3: Add Corresponding Elements

Since matrix addition involves adding corresponding elements, calculate \(C + F\) by adding corresponding elements from each matrix:\[C + F = \begin{bmatrix} 1 + 0 & 5 + 9 \ 8 + 78 & 92 + 17 \ 12 + 15 & 6 + 4 \end{bmatrix} = \begin{bmatrix} 1 & 14 \ 86 & 109 \ 27 & 10 \end{bmatrix}\]

Step 4: Verify Calculations

Double-check each addition calculation:- First row: \(1+0 = 1\), \(5+9 = 14\)- Second row: \(8+78 = 86\), \(92+17 = 109\)- Third row: \(12+15 = 27\), \(6+4 = 10\)

Key Concepts

Matrix Dimensions

When working with matrices, it's essential to understand their dimensions. The dimension of a matrix is given as "rows \( \times \) columns". This definition tells you the structure of the matrix; essentially, how many rows it has and how many columns it contains.
For example, consider matrix \( C \) and matrix \( F \) from our exercise. Both matrices have the dimensions \( 3 \times 2 \). This means:

• Each matrix has 3 rows.
• Each matrix has 2 columns.

This dimensional understanding is the foundation for performing operations like addition or multiplication on matrices. Always check the dimensions first, as this will determine the type of operations you can perform. Knowing the dimensions also helps to visualize the matrix structure, which is crucial for understanding how matrix operations work.

Compatibility for Addition

Before you dive into matrix addition, you need one crucial check: whether the matrices you are dealing with are compatible for addition.
Matrix addition is only possible when the matrices have the same dimensions. Simple rule: two matrices can only be added if they have the same number of rows and the same number of columns. Otherwise, the addition is undefined.
In our example, add matrices \( C \) and \( F \):

• Matrix \( C \) = \( 3 \times 2 \)
• Matrix \( F \) = \( 3 \times 2 \)

Both have identical dimensions, so we can proceed with adding them. If the dimensions differed, say one was \( 3 \times 2 \) and the other \( 2 \times 3 \), the addition could not be performed, as their shapes do not align for element-wise operations.

Element-wise Addition

Once you've confirmed that the matrices have the same dimensions, the next step is to add them through element-wise addition.
Element-wise addition means that you take each element from one matrix and add it to the corresponding element in the other matrix. In our case, adding matrix \( C \) to matrix \( F \) involves pairing each element in the same position across the two matrices:

• First row, first column: \( 1 + 0 = 1 \)
• First row, second column: \( 5 + 9 = 14 \)
• Second row, first column: \( 8 + 78 = 86 \)
• Second row, second column: \( 92 + 17 = 109 \)
• Third row, first column: \( 12 + 15 = 27 \)
• Third row, second column: \( 6 + 4 = 10 \)

Thus, the resulting matrix from adding \( C \) and \( F \) is:\[\begin{bmatrix}1 & 14 \86 & 109 \27 & 10 \end{bmatrix}\]This element-wise process ensures each corresponding position in the result is the sum of its counterparts in the given matrices.