Question

4 Find the transpose, minor,cofactor, & adjoint of matrix?
1 1 2
2 4 -3
3 6 -5

Answer 1

The given matrix is:
1 1 2
2 4 -3
3 6 -5

1. Transpose of a matrix:
The transpose of a matrix switches its rows and columns. Therefore, the transpose of the given matrix is:
1 2 3
1 4 6
2 -3 -5

2. Minor of a matrix:
The minor of an element in a matrix is the determinant of the submatrix formed by removing the row and column containing that element. A matrix does not have a single minor, but the minors can be found for each element individually:
- Minor of element 1: The submatrix formed by removing the row and column containing element 1 is:
4 -3
6 -5
The determinant of this 2x2 submatrix is (4 * -5) - (-3 * 6) = -14 + 18 = 4.

- Minor of element 2: The submatrix formed by removing the row and column containing element 2 is:
2 2
6 -5
The determinant of this 2x2 submatrix is (2 * -5) - (2 * 6) = -10 - 12 = -22.

- Minor of element 3: The submatrix formed by removing the row and column containing element 3 is:
2 4
6 -3
The determinant of this 2x2 submatrix is (2 * -3) - (4 * 6) = -6 - 24 = -30.

3. Cofactor of a matrix:
The cofactor of an element in a matrix is calculated by multiplying the minor of that element by (-1)^(i + j), where i is the row index and j is the column index of the element. The cofactors can be found for each element individually:
- Cofactor of element 1: (-1)^(1 + 1) * 4 = 4
- Cofactor of element 2: (-1)^(1 + 2) * -22 = 22
- Cofactor of element 3: (-1)^(1 + 3) * -30 = -30

4. Adjoint of a matrix:
The adjoint of a matrix is obtained by taking the transpose of the matrix of cofactors. Therefore, the adjoint of the given matrix is:
4 22 -30
(The transpose of the matrix of cofactors.)

Note: The adjoint of a matrix is also known as the adjugate or classical adjoint.