- Matrix addition is defined component-wise, meaning \(C_{ij} = A_{ij} + B_{ij}\).
- Identify elements \(A_{12}=9, B_{12}=0\), leading to \(C_{12} = 9 + 0 = 9\).
- Identify elements \(A_{31}=4, B_{31}=-3\), leading to \(C_{31} = 4 - 3 = 1\).
- Identify elements \(A_{22}=0, B_{22}=8\), leading to \(C_{22} = 0 + 8 = 8\).
- The result is \(C_{12} = 9, C_{31} = 1, C_{22} = 8\), written as \(\boxed{C_{12}=9, C_{31}=1, C_{22}=8}\).
Explanation
- Analysis of Matrix Addition
The matrix \(C\) is formed by adding two \(3 \times 2\) matrices, \(A\) and \(B\), component-wise. We are given:
\[A = \begin{pmatrix} 16 & 9 \ -3 & 0 \ 4 & -10 \end{pmatrix}, \quad B = \begin{pmatrix} -0.5 & 0 \ 5 & 8 \ -3 & 14 \end{pmatrix}\]
The rule for matrix addition states that the sum matrix \(C = A + B\) is obtained by adding the corresponding elements of \(A\) and \(B\), so that \(C_{ij} = A_{ij} + B_{ij}\). Our objective is to find the elements \(C_{12}\), \(C_{31}\), and \(C_{22}\).
- Identifying Elements for Calculation
To find the specific elements of matrix \(C\), we identify the corresponding elements from matrices \(A\) and \(B\):
- For \(C_{12}\) (row 1, column 2): The elements are \(A_{12} = 9\) and \(B_{12} = 0\).
- For \(C_{31}\) (row 3, column 1): The elements are \(A_{31} = 4\) and \(B_{31} = -3\).
- For \(C_{22}\) (row 2, column 2): The elements are \(A_{22} = 0\) and \(B_{22} = 8\).
- Calculating the Final Values
Now, we compute the sum of these elements:
- \(C_{12} = A_{12} + B_{12} = 9 + 0 = 9\)
- \(C_{31} = A_{31} + B_{31} = 4 + (-3) = 1\)
- \(C_{22} = A_{22} + B_{22} = 0 + 8 = 8\)
Thus, the requested elements of matrix \(C\) are \(C_{12} = 9\), \(C_{31} = 1\), and \(C_{22} = 8\).
Examples
Matrix addition is essential in data analysis and computer graphics. For example, if you are tracking the monthly profit of two different stores (represented as matrices), adding them element-by-element gives you the total profit per category for both stores combined. This helps businesses understand their cumulative performance across different sectors.