Problem 8 - 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. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{cc}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{cc}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{cc}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{cc}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{cc}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) $$ A+C $$

Step-by-Step Solution

Answer

The addition \( A+C \) is undefined due to differing dimensions.

Step 1: Determine Dimensions

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

Step 2: Check for Compatibility for Addition

In order to perform matrix addition, the matrices must 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.

Step 3: Conclusion on Addition

Since \( A \) and \( C \) do not have the same dimensions, the operation is undefined. Matrix addition cannot be performed.

Key Concepts

Understanding Matrix Dimensions

Matrices are like arrays of numbers arranged in rows and columns. The dimensions of a matrix are defined by the number of rows it has multiplied by the number of columns. For example, a matrix with 3 rows and 2 columns is denoted as a 3x2 matrix. Understanding matrix dimensions is crucial because it determines which operations can be performed.

When reading a matrix's dimensions, always specify rows first, followed by columns. For instance, matrix \( A \) in our exercise is a 2x2 matrix, while matrix \( C \) is a 3x2 matrix. This implies:

• Matrix \( A \) has 2 rows and 2 columns.
• Matrix \( C \) has 3 rows and 2 columns.

Highlighting the dimensions is the starting point to determine whether operations like addition or subtraction can proceed.

Matrix Operations: Addition and Subtraction

Matrix operations such as addition and subtraction require specific conditions to be satisfied. The primary rule is that both matrices must have identical dimensions. This means they should have the same number of rows and the same number of columns. Only then can the corresponding elements of the matrices be added or subtracted.

Consider two matrices, \( X \) and \( Y \):

• If \( X \) is a 2x3 matrix, \( Y \) must also be 2x3 to perform addition or subtraction.
• Each element in the first row of \( X \) would be added to the corresponding element in the first row of \( Y \), and so on for each row and column.

The operation would result in a new matrix of the same dimensions, in this case, 2x3. This commonality of size ensures that each element has a corresponding partner to pair with for operations.

Undefined Operations and Their Causes

In the context of matrices, an operation is referred to as "undefined" when it cannot be performed due to incompatible dimensions. Undefined operations occur when matrices do not match in size, particularly for operations like addition or subtraction that necessitate the same dimensions.

For example, matrix \( A \) is 2x2, while matrix \( C \) is 3x2. Attempting to add or subtract them is undefined because their row counts differ, despite having the same number of columns. This mismatch means there’s no way to align the rows properly for the operation.

Whenever you're unsure about whether you can add or subtract certain matrices, just remember: check their dimensions first. If they don’t match perfectly, the operation is undefined, and it cannot proceed.