Final answer:
The value of k is the dot product of vectors u and v (notated as v^Tu), which is a scalar. When the matrix (1 - uv^T) is inverted, the inverse is represented as I + (1/k)uv^T, and k is determined by solving the equation AA^{-1} = I.
Step-by-step explanation:
To find the value of k when the matrix 1 - uvT is invertible, and its inverse is given by I + (1/k)uvT, we can use the fact that a matrix multiplied by its inverse yields the identity matrix I. Let's denote the given matrix by A and its inverse by A-1. The relevant equations are:
A = 1 - uvT
A-1 = I + (1/k)uvT
To fulfill the condition AA-1 = I, we perform the multiplication and set it to the identity matrix:
(1 - uvT)(I + (1/k)uvT) = I
Let's carry out this matrix multiplication:
1 ∙ I + (1/k) ∙ uvT - uvT ∙ I - (1/k) ∙ uvTuvT = I
Simplifying and using the property that any matrix multiplied by the identity matrix I is itself, we get:
I + (1/k)uvT - uvT - (1/k)(vTu)uvT = I
The term uvT and -uvT cancel each other out and we are left with:
I - (1/k)(vTu)uvT = I
For the left side to equal the identity matrix, the matrix uvT must vanish. This implies:
(vTu)uvT = k ∙ I
Since vTu is a scalar, and the product with uvT results in a matrix proportional to the original uvT, we can infer that k equals vTu. Therefore, the value of k, in this case, is the dot product of the vectors v and u.