- Identify the dimensions of all given matrices: the first two are \(2 \times 1\), the third is \(3 \times 1\), and the fourth is \(1 \times 3\).
- Recall the rule that matrix addition requires identical dimensions.
- Compare the dimensions for each pair: only the first pair matches (\(2 \times 1\) and \(2 \times 1\)).
- Conclude that only the first matrix addition is possible, giving the result \(\boxed{\text{only the first matrix addition}}\).
Explanation
- Analyzing Matrix Dimensions
Matrix addition is only defined for two matrices that share the same dimensions, meaning they must have the same number of rows and the same number of columns. Let us analyze the dimensions of each matrix provided:
- The matrix \(\begin{bmatrix} 2 \ 3 \end{bmatrix}\) has 2 rows and 1 column, so it is a \(2 \times 1\) matrix.
- The matrix \(\begin{bmatrix} 3 \ 4 \end{bmatrix}\) has 2 rows and 1 column, so it is a \(2 \times 1\) matrix.
- The matrix \(\begin{bmatrix} 2 \ 3 \ 1 \end{bmatrix}\) has 3 rows and 1 column, so it is a \(3 \times 1\) matrix.
- The matrix \(\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\) has 1 row and 3 columns, so it is a \(1 \times 3\) matrix.
- Evaluating Matrix Addition Problems
Now we evaluate each of the three given addition problems based on the condition that dimensions must be identical:
- \(\begin{bmatrix} 2 \ 3 \end{bmatrix} + \begin{bmatrix} 3 \ 4 \end{bmatrix}\): Both are \(2 \times 1\) matrices. Since the dimensions match, this operation is possible. The result is \(\begin{bmatrix} 5 \ 7 \end{bmatrix}\).
- \(\begin{bmatrix} 2 \ 3 \ 1 \end{bmatrix} + \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\): The first is a \(3 \times 1\) matrix and the second is a \(1 \times 3\) matrix. Since these dimensions are different (\(3 \times 1
eq 1 \times 3\)), this operation is not possible.
- \(\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} + \begin{bmatrix} 2 \ 3 \ 1 \end{bmatrix}\): The first is a \(1 \times 3\) matrix and the second is a \(3 \times 1\) matrix. Since these dimensions are different, this operation is not possible.
- Final Conclusion
Based on our analysis, only the first expression \(\begin{bmatrix} 2 \ 3 \end{bmatrix} + \begin{bmatrix} 3 \ 4 \end{bmatrix}\) is possible. The final answer is \(\boxed{\text{only the first matrix addition}}\).
Examples
Matrix addition is essential in data processing and computer science. For example, if a store tracks inventory across two different days using matrices, they can only sum those matrices if the rows and columns represent the same products and categories. If Day 1 has 3 items listed as a column vector and Day 2 has a row vector of 3 items, you cannot add them directly; you would first need to ensure both are in the same orientation to perform the update. Understanding dimensions prevents errors when scaling these operations across large datasets.