- Perform component-wise addition on the given matrices: \(\begin{bmatrix} n-2 & 6 \ -3 & m-5 \end{bmatrix} = \begin{bmatrix} 10 & 6 \ -3 & 40 \end{bmatrix}\).
- Extract the equations for \(n\) and \(m\): \(n-2=10\) and \(m-5=40\).
- Solve the equations: \(n=12\) and \(m=45\).
- The final answer is \(\boxed{m=45, n=12}\).
Explanation
- Understanding Matrix Addition
The problem asks us to determine the values of \(m\) and \(n\) that satisfy the given matrix equation:
\[\left[\begin{array}{cc} n-1 & 6 \ -19 & m+3 \end{array}\right] + \left[\begin{array}{cc} -1 & 0 \ 16 & -8 \end{array}\right] = \left[\begin{array}{cc} 10 & 6 \ -3 & 40 \end{array}\right]\]
Matrix addition is performed by adding corresponding elements of the matrices. This results in a system of four linear equations corresponding to the entries in the \(2 \times 2\) matrix. We will extract each equation and solve for the variables \(n\) and \(m\).
- Setting Up the Equations
We add the components of the two matrices on the left side:
\(\left[\begin{array}{cc} (n-1)+(-1) & 6+0 \ -19+16 & (m+3)+(-8) \end{array}\right] = \left[\begin{array}{cc} 10 & 6 \ -3 & 40 \end{array}\right]\)
Now, we set each corresponding component equal to the component in the resulting matrix:
- For the first row, first column: \(n - 1 - 1 = 10\)
- For the first row, second column: \(6 + 0 = 6\) (This is consistent)
- For the second row, first column: \(-19 + 16 = -3\) (This is consistent)
- For the second row, second column: \(m + 3 - 8 = 40\)
- Solving for Variables and Final Selection
Now we solve the equations for \(n\) and \(m\):
For \(n\):
\(n - 2 = 10\)
Adding 2 to both sides gives:
\(n = 12\)
For \(m\):
\(m - 5 = 40\)
Adding 5 to both sides gives:
\(m = 45\)
Thus, we found \(n = 12\) and \(m = 45\). Comparing this result with the given options, we identify the correct choice.
Examples
Matrix addition is widely used in computer graphics to transform objects, such as shifting the position of an image or a 3D model. For instance, if you have a set of coordinates representing the vertices of a shape and you want to move the entire object, you can add a displacement matrix to the coordinate matrix to update the position of every point simultaneously. This method is fundamental for efficiency in game development and animation software.