Question

The eigenvalue of matricx(111 111 111)

Answer 1

The eigenvalue of the matrix
`\( \begin{bmatrix} 111 & 111 & 111 \end{bmatrix} \)` is obtained by solving the characteristic equation
`\( \text{det}(A - \lambda I) = 0 \), where \( A \)` is the matrix, \( \lambda \) is the eigenvalue, and
`\( I \)` is the identity matrix.

Step-by-step explanation:

The given matrix is
`\( A = \begin{bmatrix} 111 & 111 & 111 \end{bmatrix} \)`. To find the eigenvalue
`\( \lambda \), we subtract \( \lambda \)` times the identity matrix from
`\( A \):`

`\[ A - \lambda I = \begin{bmatrix} 111-\lambda & 111 & 111 \\ 111 & 111-\lambda & 111 \\ 111 & 111 & 111-\lambda \end{bmatrix} \]`

Next, we calculate the determinant of this matrix and set it equal to zero:

`\[ \text{det}(A - \lambda I) = (111-\lambda)^3 \]`

Setting this expression equal to zero and solving for
`\( \lambda \)` gives:

`\[ (111-\lambda)^3 = 0 \]`

Taking the cube root of both sides, we find that
`\( \lambda = 111 \).`Therefore, the eigenvalue of the matrix
`\( \begin{bmatrix} 111 & 111 & 111 \end{bmatrix} \)` is 111.

In conclusion, the detailed calculation involved forming the characteristic equation, finding the determinant, setting it equal to zero, and solving for
`\( \lambda \).`The solution
`\( \lambda = 111 \)` represents the eigenvalue of the given matrix.