Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$\left[\begin{array}{rrr}-8 & 4 & 0 \\ 2 & 5 & 0\end{array}\right]+\left[\begin{array}{ll}6 & 3 \\ 8 & 9\end{array}\right]$$

Step-by-Step Solution

Answer

The matrices cannot be added as they have different dimensions.

Step 1: Check Matrix Dimensions

The first step is to verify the dimensions of the two matrices involved in the operation. The first matrix, \( A \), is a 2x3 matrix:\[A = \begin{bmatrix} -8 & 4 & 0 \ 2 & 5 & 0 \end{bmatrix}\]The second matrix, \( B \), is a 2x2 matrix:\[B = \begin{bmatrix} 6 & 3 \ 8 & 9 \end{bmatrix}\]To add two matrices, they must have the same dimensions. Compare their sizes to check if the operation is possible.

Step 2: Determine Compatibility for Addition

Analyzing the dimensions: Matrix \( A \) is 2x3, while Matrix \( B \) is 2x2. Since they do not have the same dimensions, these matrices cannot be added together. Matrix addition is defined only when both matrices are of identical dimensions.

Step 3: Conclusion

Since the matrices \( A \) and \( B \) are not of the same size, it is impossible to perform the addition. The operation cannot be completed as matrix addition requires both matrices to have identical dimensions.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra where you add two matrices together element by element. However, this operation is only valid under specific conditions. To perform matrix addition, each matrix must have the same dimensions. In other words, both matrices must have the same number of rows and columns. Addition is performed by adding the corresponding elements from each matrix.
For example, if you have two matrices:

• Matrix C: \[\begin{bmatrix} 1 & 2 & 3 \4 & 5 & 6 \end{bmatrix}\]
• Matrix D:\[\begin{bmatrix}7 & 8 & 9 \10 & 11 & 12\end{bmatrix}\]

You can add them to form a new matrix:\[\begin{bmatrix}1+7 & 2+8 & 3+9 \4+10 & 5+11 & 6+12\end{bmatrix}\]This gives us the result:\[\begin{bmatrix}8 & 10 & 12 \14 & 16 & 18\end{bmatrix}\]Matrix addition is a straightforward process, but it requires the matrices to be of the same size. This ensures that each element in the first matrix has a corresponding element in the second matrix to add to.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing any matrix operations, such as addition, subtraction, and multiplication. The dimensions of a matrix are given in the form of rows by columns, often written as "m x n" where "m" is the number of rows and "n" is the number of columns.
For example, if a matrix "E" is described as having dimensions 3x2, it means that Matrix E has 3 rows and 2 columns.

• Matrix:\[\begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22} \a_{31} & a_{32}\end{bmatrix}\]

When checking the dimensions of the matrices, it's important to ensure compatibility for operations you want to perform. If you try to add two matrices with different dimensions, as seen in the example exercise, it will not be possible because each element in one matrix must match one in the other. Thus, knowing matrix dimensions helps in determining what operations can be performed.

Matrix Compatibility

Matrix compatibility refers to the requirement that matrices must fulfill specific conditions to undergo certain operations. Not all matrices can be added or multiplied together. For addition, the number of rows and columns in both matrices must match exactly.
In matrix multiplication, however, compatibility depends on the inner dimensions matching. This means if you have one matrix of "p x q" and a second of "q x r," they can be multiplied because the number of columns of the first matrix equals the number of rows of the second matrix.

• For addition: Matrices must have the same dimensions.
• For multiplication: The number of columns in the first matrix must equal the number of rows in the second.

If matrices don't meet these criteria, the operations can't be performed. Always check matrix compatibility before attempting any operations to avoid errors. Proper understanding of this concept will save time and prevent mistakes in mathematical computations.