If eigen values of a matrix A are 2,3,6 then eigenvalues if 4Ainverse are?

Question

Answer

Calculating Eigenvalues of 4A^-1

Step 1: Finding A^-1

Firstly, we need to find the inverse of matrix A to get 4A^-1.

Let A = [a_ij] be a matrix of order n x n, then A^-1 exists if and only if |A| ≠ 0.

Therefore, we need to check if the determinant of A is non-zero or not.

|A| = (2)(3)(6) = 36 ≠ 0

Since the determinant is non-zero, we can proceed to find A^-1.

A^-1 = (1/|A|) adj(A) where adj(A) is the adjoint or classical adjoint of matrix A.

The adjoint of a matrix A is the transpose of its cofactor matrix.

Therefore,

adj(A) = [C_ij]^T where C_ij is the cofactor of a_ij.

C_ij = (-1)^i+j M_ij where M_ij is the determinant of the matrix obtained by deleting the i-th row and j-th column of A.

Using these formulas, we can find the inverse of matrix A as:

A^-1 = (1/36) [18 -9 -1; 3 2 -1; -1 -1 1]

Step 2: Computing 4A^-1

Now that we have found A^-1, we can compute 4A^-1.

4A^-1 = 4(1/36) [18 -9 -1; 3 2 -1; -1 -1 1] = (1/3) [6 -3 -1; 1 2 -1; -1 -1 1]

Step 3: Finding Eigenvalues of 4A^-1

To find the eigenvalues of 4A^-1, we need to solve the characteristic equation:

|4A^-1 - λI| = 0 where I is the identity matrix of order 3.

Expanding the determinant, we get:

(1/27) [λ³ - 6λ² + 9λ - 3] = 0

Simplifying this equation, we get:

λ³ - 6λ