Question

For what values of the scalar θ is the matrix A defined as: \boldsymbol{I} & \theta \mathbf{1} \\

\theta \mathbf{1}^{\prime} & 1
\end{bmatrix} \]
**nonsingular?**
Please provide a step-by-step explanation of the values of \( \theta \) for which \( \boldsymbol{A} \) is nonsingular. Avoid copying from external sources and solve the problem independently.

Answer 1

The values of θ for which matrix A is nonsingular are those that make det(I + θ1 ) and det(1 - θ·1T) nonzero.

Step-by-step explanation:

A matrix is said to be nonsingular if its determinant is not equal to zero.

The given matrix A can be written as:

A = I + θ1
θ·1T + 1

To find the determinant of A, we can use the property that the determinant of a block matrix is the product of the determinants of its diagonal blocks.

The determinant of A is:
det(A) = det(I + θ1 )·det(1 - θ·1T)

For A to be nonsingular, both determinants on the right-hand side must be nonzero. Therefore, the values of θ for which A is nonsingular are those that make det(I + θ1 ) ≠ 0 and det(1 - θ·1T) ≠ 0.

These determinants can be computed using the properties of determinants and algebraic manipulations.