Problem 11 - Precalculus

Question

Use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\ -\frac{4}{3} & 0 \\\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ E+D $$

Step-by-Step Solution

Answer

The operation \(E + D\) is undefined since the matrices have different dimensions.

Step 1: Check Compatibility for Addition

To add two matrices, they must have the same dimensions, which means the same number of rows and columns. Matrix \(E\) is a 3x3 matrix, and matrix \(D\) is a 3x2 matrix. Since they have different dimensions, the matrices cannot be added together.

Step 2: Conclusion on Matrix Addition

Since the matrices \(E\) and \(D\) do not have the same dimensions, the operation \(E + D\) is undefined.

Key Concepts

Matrix Addition

Matrix addition is a straightforward process akin to adding numbers, but it comes with its own set of rules. When adding matrices, each matrix must be of the same size, meaning they have to have the same number of rows and columns. If you imagine matrices as grids of numbers, for two matrices to be added, they must be the same size grid. Here's how you proceed with matrix addition:

• Align the matrices by rows and columns.
• Add corresponding elements (elements that are in the same position).
• Write the sum in a new matrix.

Let's illustrate this further. Given two matrices, if matrix A has an element in the first row, second column, that position is added with the same position from matrix B. It is essential to remember that the sum of these positions will form the same position in the resulting matrix.

Matrix Dimensions

Matrix dimensions are a fundamental aspect when dealing with matrix operations. They define the size of the matrix. A matrix's size is represented by its number of rows by its number of columns. So, for a matrix with 3 rows and 2 columns, we denote it as a 3x2 matrix. Understanding these dimensions is crucial for anyone working with matrices because they determine compatibility for various operations, like addition and multiplication.

• Rows: Horizontal lines of numbers in the matrix.
• Columns: Vertical lines of numbers in the matrix.

Being precise with dimensions helps ensure operations are executed correctly and guards against errors.

Incompatibility in Matrices

Incompatibility arises in matrix operations when certain conditions are not met, preventing the operation from being completed. In matrix addition, this incompatibility occurs when the matrices involved do not share the same dimensions. If two matrices do not have the same number of rows and columns, the operation becomes undefined. For instance, attempting to add a 3x3 matrix to a 3x2 matrix will result in an undefined operation as their dimensions do not align perfectly. Always check dimensions before attempting to add, subtract, or perform similar operations on matrices. This step ensures that the operation can be performed smoothly without computational errors. Guarding against such incompatibility is key to working effectively with matrices.