Let A be an orthogonal non-singular matrix of order 'n', then the determinant of matrix 'A-I', i.e., |A-I| (where I is identity matrix) is?
To find the determinant of the matrix 'A-I' where A is an orthogonal non-singular matrix of order 'n', we need to calculate the eigenvalues of 'A'.
Step 1: Find the eigenvalues of 'A'
- An orthogonal matrix has the property that its eigenvalues have a magnitude of 1. This means that all the eigenvalues of 'A' will be either 1 or -1.
Step 2: Find the determinant of 'A-I'
- To get the determinant of 'A-I', we replace every element on the diagonal of 'A' with the difference between that element and 1.
For example, if 'A' is a 3x3 matrix:
|A-I| = |a11-1 a12 a13|
|a21 a22-1 a23|
|a31 a32 a33-1|
Step 3: Calculate the determinant
- The determinant of a matrix can be found by multiplying the eigenvalues of the matrix. Since 'A' is orthogonal non-singular, all the eigenvalues are either 1 or -1, so the determinant will be the product of these eigenvalues.
For example, if the eigenvalues of 'A' are {1, -1, 1}, then the determinant would be 1*(-1)*1 = -1.
Therefore, the determinant of matrix 'A-I' (where A is an orthogonal non-singular matrix of order 'n') will be either 1 or -1, depending on the eigenvalues of 'A'.
To find the determinant of the matrix A-I, where A is an orthogonal non-singular matrix of order 'n', we can use the following steps:
1. Start with the matrix A-I.
2. Subtract the identity matrix I from A by subtracting the corresponding elements in each row.
3. The resulting matrix will have the form:
| a11 - 1 a12 a13 ... a1n |
| a21 a22 - 1 a23 ... a2n |
| a31 a32 a33 - 1 ... a3n |
| ... |
| an1 an2 an3 ... ann - 1 |
4. Since A is orthogonal, the columns of A are orthogonal unit vectors. Therefore, the diagonal elements of A are either 1 or -1.
5. Subtracting 1 from the diagonal elements of A will result in elements that are either 0 or -2.
6. Taking the determinant of the matrix A-I is equivalent to calculating the product of the diagonal elements.
7. Since the diagonal elements are either 0 or -2, the determinant will be 0 if there is at least one 0 on the diagonal, otherwise it will be (-2)^n.
So, the determinant of the matrix A-I, i.e., |A-I|, is 0 if there is at least one 0 on the diagonal, otherwise it is (-2)^n.