Calculating adj A and A^-1:
1. Finding the matrix A:
The given system of equations can be written in matrix form as:
A = [3 2 -5; 0 3 4; -1 2 1]
2. Finding the determinant of A:
det(A) = 3(3*1 - 2*4) - 2(0*1 - 5*4) + (-5*(0*2 - 3*1))
det(A) = 3(3-8) - 2(0-20) + (-5(0-3))
det(A) = 3*(-5) - 2*(-20) + (-5*(-3))
det(A) = -15 + 40 + 15
det(A) = 40
3. Finding the adjoint of A:
The adjoint of a matrix is the transpose of the cofactor matrix.
Cofactor matrix = [3 -7 -3; 8 -5 -6; 3 -9 -3]
Adjoint matrix = [3 8 3; -7 -5 -9; -3 -6 -3]
4. Calculating adj A:
adj(A) = 1/det(A) * adjoint of A
adj(A) = 1/40 * [3 8 3; -7 -5 -9; -3 -6 -3]
adj(A) = [3/40 8/40 3/40; -7/40 -5/40 -9/40; -3/40 -6/40 -3/40]
5. Finding the inverse of A:
A^-1 = adj(A) / det(A)
A^-1 = [3/40 8/40 3/40; -7/40 -5/40 -9/40; -3/40 -6/40 -3/40] / 40
A^-1 = [3/160 1/5 3/160; -7/160 -1/8 -9/160; -3/160 -3/20 -3/160]
Therefore, the adjoint of A is [3/40 8/40 3/40; -7/40 -5/40 -9/40; -3/40 -6/40 -3/40] and the inverse of A is [3/160 1/5 3/160; -7/160 -1/8 -9/160; -3/160 -3/20 -3/160].