- Identify the position of variables \(a, b, c\) within the matrix: \(a\) at \((2, 1)\), \(b\) at \((3, 2)\), and \(c\) at \((4, 2)\).
- Perform component-wise addition for each variable: \(a = -10 + (-4) = -14\), \(b = 3 + (-12) = -9\), and \(c = -9 + (-3) = -12\).
- Final values are \(a = \boxed{-14}\), \(b = \boxed{-9}\), and \(c = \boxed{-12}\).
Explanation
- Understanding Matrix Addition
Matrix addition is performed by adding the corresponding elements of two matrices. Given two matrices \(A\) and \(B\) of the same dimensions, the sum \(A+B\) is a matrix where each entry \((A+B)_{ij}\) is calculated as \(A_{ij} + B_{ij}\). We are given:
\[ \begin{bmatrix} 4 & 1 & -5 \ -10 & 2 & 13 \ 3 & 3 & 0 \ -8 & -9 & 6 \end{bmatrix} + \begin{bmatrix} -8 & 0 & 6 \ -4 & 5 & -1 \ 15 & -12 & 3 \ 0 & -3 & 11 \end{bmatrix} = \begin{bmatrix} -4 & 1 & 1 \ a & 7 & 12 \ 18 & b & 3 \ -8 & c & 17 \end{bmatrix} \]
Our goal is to isolate the variables \(a, b,\) and \(c\) by performing the addition on the specific components where they are located.
- Calculating the Variables
To find the value of \(a\), we look at the element in the second row, first column of the resulting matrix. By adding the corresponding elements from the two given matrices, we have:
\[ a = (-10) + (-4) = -14 \]
To find the value of \(b\), we look at the element in the third row, second column of the resulting matrix:
\[ b = 3 + (-12) = -9 \]
To find the value of \(c\), we look at the element in the fourth row, second column of the resulting matrix:
\[ c = (-9) + (-3) = -12 \]
Thus, we have successfully determined the variables by performing element-wise addition.
- Conclusion and Final Results
By performing the matrix addition, we have identified that:
\(a = -14\),
\(b = -9\),
\(c = -12\).
These values satisfy the matrix equation given in the problem statement. Therefore, the final values are \(a = -14\), \(b = -9\), and \(c = -12\).
Examples
Matrix addition is a fundamental tool used in computer graphics and data processing. For instance, if you have a set of coordinates representing a 2D object and you want to move (translate) the entire object on a grid, you can represent the movement as a matrix of displacement vectors. By adding the displacement matrix to your original coordinate matrix, you efficiently calculate the new position of every point on your object simultaneously. This is how digital characters move smoothly across your screen in video games.