Final answer:
To find the minors and cofactors of a matrix, we calculate the determinant of submatrices obtained by removing rows and columns. The sign of each determinant determines the sign of the corresponding cofactor. The minors and cofactors of the given matrix are 4, -51, 77, and -19.
Step-by-step explanation:
To find all the minors and cofactors of a matrix, we need to first understand what minors and cofactors are. The element's minor in a matrix is the determinant of the submatrix obtained by discarding the row and column containing the element. The cofactor of an element is the signed minor, where the sign is determined by the position of the element in the matrix.
In the given matrix, the minors and cofactors can be found as follows:
- Minor of element 2 is the determinant of the submatrix [4]. Its value is 4.
- Minor of element 1 is the determinant of the submatrix [3, 20; -2, -9]. Its value is 51.
- Minor of element 3 is the determinant of the submatrix [1, 20; -7, -9]. Its value is 77.
- Minor of element 4 is the determinant of the submatrix [1, 3; -7, -2]. Its value is -19.
The cofactors can be calculated by multiplying each minor by the corresponding sign factor, which is determined by the position of the element. In this case, the sign factors are +1, -1, +1, and -1 for elements 2, 1, 3, and 4 respectively.
Therefore, the cofactors are 4, -51, 77, and -19.