- Identify the corresponding elements in the given \(2 \times 2\) matrices.
- Perform element-wise addition: \(3 + 6 = 9\), \(9 + 0 = 9\), \(5 + (-8) = -3\), and \(-2 + 4 = 2\).
- Combine the results into the matrix \(\begin{bmatrix} 9 & 9 \ -3 & 2 \end{bmatrix}\).
- Match the result with the second provided option.
- The final answer is \(\boxed{\begin{bmatrix} 9 & 9 \ -3 & 2 \end{bmatrix}}\).
Explanation
- Understanding Matrix Addition
We are given two \(2 \times 2\) matrices:
\[ A = \begin{bmatrix} 3 & 9 \ 5 & -2 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & 0 \ -8 & 4 \end{bmatrix} \]
To find the sum of these matrices, we add the corresponding elements from each matrix. Specifically, for two \(2 \times 2\) matrices \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\) and \(\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\), their sum is defined as:
\[ \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \]
- Performing the Addition
We identify the corresponding elements and calculate their sums:
- Top-left element: \(3 + 6 = 9\)
- Top-right element: \(9 + 0 = 9\)
- Bottom-left element: \(5 + (-8) = -3\)
- Bottom-right element: \(-2 + 4 = 2\)
Combining these values into the resulting matrix gives:
\[ \begin{bmatrix} 9 & 9 \ -3 & 2 \end{bmatrix} \]
- Finalizing the Solution
Comparing our result \(\begin{bmatrix} 9 & 9 \ -3 & 2 \end{bmatrix}\) with the given options:
- Option 1: \(\begin{bmatrix} -3 & 9 \ 13 & -6 \end{bmatrix}\)
- Option 2: \(\begin{bmatrix} 9 & 9 \ -3 & 2 \end{bmatrix}\)
- Option 3: \(\begin{bmatrix} -3 & 2 \ 9 & 9 \end{bmatrix}\)
- Option 4: \(\begin{bmatrix} 9 & 9 \ 3 & 2 \end{bmatrix}\)
Our calculated result matches the second option. The sum is \(\boxed{\begin{bmatrix} 9 & 9 \ -3 & 2 \end{bmatrix}}\).
Examples
Matrix addition is essential in computer science for processing images, where each pixel is represented as a value in a matrix. For example, if you want to increase the brightness of an image, you can add a constant matrix to the matrix representing the pixel intensities. Similarly, in game development, matrices are used to transform the position of 3D objects, and adding matrices can help translate or move these objects within a 3D coordinate system. Understanding how to correctly add matrices is a fundamental skill for data science and computer graphics.