Question: Suppose that the eigenvalues of matrix A are 1, 2, and 4. The determinant of (A-1)T is _______________. Correct answer is '0.125'. Can you explain this answer?

Answer:

To find the determinant of the matrix (A-1)T, we first need to understand a few concepts and properties related to determinants and eigenvalues.

Eigenvalues:
Eigenvalues are the values λ for which the equation Av = λv holds true, where A is the matrix and v is the eigenvector.

Determinant:
The determinant of a square matrix gives us information about the matrix's properties, such as invertibility and the scaling factor of the linear transformation represented by the matrix.

Properties of Determinants:
1. If A is a square matrix and k is a scalar, then det(kA) = k^n * det(A), where n is the order/dimension of the matrix.
2. If A and B are square matrices of the same order, then det(AB) = det(A) * det(B).
3. If A is a square matrix and B is its transpose, then det(A) = det(B).

Now, let's proceed with finding the determinant of (A-1)T.

Step 1: Find the determinant of A.
Since the eigenvalues of matrix A are given as 1, 2, and 4, we can write A as a diagonal matrix with these eigenvalues on the diagonal.

A =
| 1 0 0 |
| 0 2 0 |
| 0 0 4 |

The determinant of A is the product of its eigenvalues: det(A) = 1 * 2 * 4 = 8.

Step 2: Subtract 1 from each eigenvalue of A and form a new matrix (A-1).
(A-1) =
| 0 -1 -1 |
| -1 1 -1 |
| -1 -1 3 |

Step 3: Take the transpose of (A-1).
(A-1)T =
| 0 -1 -1 |
| -1 1 -1 |
| -1 -1 3 |

Step 4: Find the determinant of (A-1)T.
Using the properties of determinants, we can calculate the determinant of (A-1)T as follows:

det((A-1)T) = det(A-1)

Since (A-1) is a 3x3 matrix, we can expand the determinant using the first row:

det((A-1)T) = 0 * det(minor) - (-1) * det(minor) + (-1) * det(minor)

Simplifying further, we get:

det((A-1)T) = det(minor) + det(minor) - det(minor)
= 2 * det(minor)

Step 5: Find the determinant of the minor.
To find the determinant of the minor, we can expand it using the first row:

det(minor) = 1 * det