To find the value of a+b+c+d, we need to equate the corresponding elements of the given matrices. The given equation is:
Let's multiply the matrices on the right-hand side first. The multiplication of two matrices is defined as follows:
Now let's distribute the scalar 2 into the first matrix on the left-hand side and add it to the second matrix:
Equating the corresponding elements of the matrices gives us the following system of equations:1.
2.
3.
4.
Solving these equations gives us:1.
2.
3.
4. \(b = cd+10 \rightarrow b = (-5)(-1)+10 = 15\)Now, we calculate the sum of a, b, c, and d:
However, this result does not match any of the answer choices given, indicating there might have been a mistake in the calculation. Let's check the equations again to make sure:For equation 2, correcting the algebra gives us
, which is correct.For equation 3, correcting the algebra gives us
, which is correct.Re-evaluating equation 4 with the correct values of c and d gives us \(b = (-5)(-1)+10 = 5+10 = 15\), which is also correct.Now let's recalculate the sum with the corrected values:
This still doesn't match any of the given options. Given that we have thoroughly checked the calculations and the solution does not fit any of the multiple-choice answers, the question might be flawed or the options might be incorrect.Therefore, we conclude with the output that no further answer is needed due to inconsistency in the question or provided options.