Problem 8 - College Algebra

Question

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

Step-by-Step Solution

Answer

Matrix addition is undefined due to different dimensions.

Step 1: Determine Matrix Dimensions

First, identify the dimensions of the given matrices. Matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(C\) is a \(3 \times 2\) matrix.

Step 2: Check for Compatibility

Matrix addition requires both matrices to have the same dimensions. Since matrix \(A\) is \(2 \times 2\) and matrix \(C\) is \(3 \times 2\), they do not have the same dimensions and cannot be added.

Step 3: Declare the Operation Undefined

Since the matrices have different dimensions, matrix addition is undefined in this case.

Key Concepts

Matrix Addition

Matrix addition is a straightforward operation in linear algebra. It involves adding corresponding elements of two matrices to produce a new matrix. Each element in the resulting matrix is the sum of the elements from the same position in the two matrices being added.
However, it's crucial to remember that addition can only be performed if both matrices have the same dimensions. This means they must have the same number of rows and columns. This requirement ensures each element has a corresponding partner to add to, resulting in a well-defined output matrix.

• Step 1: Identify the dimensions of each matrix.
• Step 2: Check if the dimensions are identical.
• Step 3: Perform the addition, element by element.

If any of the above steps reveal that the dimensions are not the same, the matrices cannot be added together. Thus, the operation is considered undefined.

Matrix Subtraction

Much like matrix addition, matrix subtraction is an operation where you subtract corresponding elements from each other. It also requires the matrices to have the same dimensions.
In other words, to perform matrix subtraction:

• Each matrix must have the same number of rows and columns.
• Subtract the element from one matrix from the corresponding element in the other matrix.

Similar to addition, if the dimensions do not match, subtraction cannot be performed, and the operation becomes undefined.

Matrix Dimensions

Understanding matrix dimensions is critical in matrix operations. Matrix dimensions are expressed as "rows by columns," for example, a 2x3 matrix has 2 rows and 3 columns.
Identifying dimensions is the first step in determining whether operations like addition or subtraction are possible:

• Count the number of rows.
• Count the number of columns.

This seemingly simple task is key for verifying compatibility and proceeding with any further calculations. Dimensions dictate compatibility in matrix operations.

Compatibility in Matrix Operations

Compatibility in matrix operations is fundamental to whether operations like addition or subtraction can be performed. Specifically:

• Matrices must have the same dimensions for addition and subtraction: identical rows and columns are needed.
• Lack of matching dimensions means the operation is undefined.

In practice, this means the numbers that form the matrix (its elements) must line up perfectly between the two matrices you're considering for the operation. This alignment is necessary so that each element has a direct counterpart to interact with.
Whenever verifying compatibility, always start by checking dimensions, ensuring both matrices have matched rows and columns. This step is non-negotiable in successful matrix mathematics operations.