Problem 10 - College Algebra

Question

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

Step-by-Step Solution

Answer

C + F = \( \begin{bmatrix} 1 & 14 \\ 86 & 109 \\ 27 & 10 \end{bmatrix} \)

Step 1: Check Dimensions

To add two matrices, they must have the same dimensions. First, check the dimensions of matrices \(C\) and \(F\). Matrix \(C\) is 3x2 (3 rows and 2 columns) and matrix \(F\) is also 3x2 (3 rows and 2 columns). Since their dimensions match, we can proceed with the addition.

Step 2: Add Corresponding Elements

Add the corresponding elements from matrices \(C\) and \(F\):- 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: \(6 + 4 = 10\)Place these sums into a new matrix.

Step 3: Write and Verify the Resulting Matrix

The resulting matrix from adding matrices \(C\) and \(F\) is:\[C+F = \begin{bmatrix} 1 & 14 \ 86 & 109 \ 27 & 10 \end{bmatrix}\]Check the work for accuracy by revisiting each calculation and confirming that each corresponding element was added correctly.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. It involves adding two matrices by combining their elements.
To perform matrix addition, the two matrices must have the same dimensions. This means that they should have the same number of rows and columns. If this condition is met, we can proceed to add the corresponding elements of each matrix.
For instance, if we have two matrices, matrix A and matrix B, with dimensions 2x2, their elements are added as follows:

• Add the element in the first row and first column of matrix A to the element in the first row and first column of matrix B.
• Continue this process for each corresponding element across the rows and columns.

Matrix addition is useful when dealing with systems of equations or transformations, as it allows for the combination of components in a straightforward manner.

Dimension Check

Before adding two matrices, it's crucial to verify that they have the same dimensions. The dimension of a matrix is described by the number of rows and columns it contains.
For example:

• A matrix is described as "3x2" if it has 3 rows and 2 columns.
• Another matrix of "3x2" dimensions can be added to the first one.

When dimensions match, you can proceed with matrix addition.
However, if they do not match, the operation is undefined, meaning it cannot be performed.
Checking dimensions is the first step in any matrix operation to ensure that the procedure is possible and to avoid calculation errors later on.

Corresponding Elements

The heart of matrix addition lies in the concept of corresponding elements. When matrices with identical dimensions are added, each element in the resultant matrix is formed by summing the corresponding elements from the two matrices.
To understand this better:

• Consider two matrices, each of dimension 2x3. The element in the first row, first column of the resulting matrix is derived from the addition of the first row, first column elements of the two matrices.
• This process is repeated for each element in the matrices.

Every position in the new matrix directly corresponds to a position in the original matrices.
By carefully matching these positions, you ensure that the process is accurate and that each element has been correctly computed.
Revising each calculation helps affirm the integrity of your resulting matrix.