Solution:
The problem at hand is to find the values of λ for which the determinant of Matrix A equals zero, where Matrix A is as follows:
A = λ - 7 0 9
0 λ + 7 0
1 1 1
```
To find the values of λ, the determinant of Matrix A must first be calculated, then equated to zero and solved for λ. Let's calculate the determinant:
Given our 3x3 matrix:
````
A = (λ - 7) 0 9
0 (λ + 7) 0
1 1 1
````
The determinant of A can be computed using the formula for 3x3 matrices:
```
det(A) = (λ - 7)*[(λ+7)*1 - 0*1] - 0*[0 - 0] + 9*[0 - 0]
Working that out, we find:
det(A) = (λ - 7)*(λ+7) = λ^2 - 9*λ - 112
```
Setting the determinant to zero to solve for λ gives the following:
λ^2 - 9λ - 112 = 0
Solving this quadratic equation for λ, we find that the values of λ that satisfy det(A) = 0 are λ = -7 and λ = 16.
Therefore, the matrix A is singular (that is, det(A) = 0) when λ = -7 or λ = 16. Out of these two solutions, the greater value is λ = 16.