Final answer:
To solve the given linear equation using the formula x∧ = (Aᵗ A)⁻¹b, we need to find the values of A and b. From the equation 25x + 5y + z = 106.8, we can determine the coefficient matrix A and the constant vector b. By substituting A, (Aᵗ A)⁻¹, and b into the formula x∧ = (Aᵗ A)⁻¹b, we can find the solution vector x.
Step-by-step explanation:
To solve the given linear equation using the formula x = (AṀ A)⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻b, we need to find the values of A and b.
From the equation 25x + 5y + z = 106.8, we can see that the coefficient matrix A is:
A = [[25, 5, 1]].
And the constant vector b is:
b = [[106.8]].
Now, we can calculate AṀ A and its inverse:
AṀ A = [[25, 5, 1], [5, 1, 0], [1, 0, 0]]
(AṀ A)⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻ = [[-0.04, 0.12, -0.08], [0.12, -0.36, 0.24], [-0.08, 0.24, -0.16]]
Finally, we can substitute A, (AṀ A)⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻, and b into the formula x = (AṀ A)⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻b to find the solution vector x.