Question
D) None IV. Work out: show all the necessary steps clearly (2.5pts) Solve the following linear system using Cramer's Rule x-y+2z=7 3x+4y-5z=-5 2x-y+3z=12 Find the eigenvalues and the corresponding eigenvectors of A=[[3,2],[3,-2]] Find a linear transformation T:R^(2)rarrR^(3) such that T(1,2)=(3,-1,5) and T(0,1)= (2,1,-1) Evaluate the cofactor of [[2,3,4],[3,2,1],[1,1,-2]] . int7bi tros -1,2],[4,-1],[1,2])([x],[4],[z])=([7],[-1],[12])
Answer
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(140 Votes)
Lester
Master · Tutor for 5 years
Answer
1. Apply the Cramer’s rule with:A={{1, -1, 2}, {3, 4, -5}, {2, -1, 3}}, B={{7, -1, 2}, {7,3, -5}, {7, -1, 3}}, C={{1, 7, 2}, {3, -5, -5}, {2, 12, 3}}, D={{1, -1, 7}, {3, 4, -5}, {2, -1, 12}}, x = Real part of determinant B / determinant A= y = Real part of determinant of C /determinant A= z = Real part of determinant D / determinant A.2.Characteristic equation of A is colve(det(A-In,lambda),lambda). Finding the null space will return us eigenvectors.3.Filtered transformation coefficients are Real[{a}, {b}, {c}, {d}, {e}, {f} matching system solutions. 4.Given that our matrix is A=[[2,3,4], [3,2,1], [1,1, -2]], the cofactor_value_of_specific_element can be calculated like follow: Adjugate=A.copy().T, statement: after last added element Adjugate = Find the simplified augmented_T value in Matrix format.
Explanation
1. Solving linear equation. Using Cramer's rule to solve a system of linear equations involves performing operations on the system's coefficient matrix and on matrices obtained from it by replacing one of its column by the column vector of constant terms. You find the so-called Cramer's determinants.Cramer’s Rule involves solving the given system of equations by following these steps:Firstly, we need to find the determinant of matrix A (original coefficients matrix), then substitute each column of A with the values after the equations equal sign, accordingly creating B, C, and D. By deriving each determinant Det.The solutions can be written as x = Det(B) / Det(A), y = Det(C) / Det(A), and z = Det(D) / Det(A). 2. To find eigenvalues we must solve the characteristic equation. This involves solving a polynomial equation. The number λ is an eigenvalue of a given matrix A exactly when the computation determinant (APIn–λAPIn) equates to zero.The eigenvectors corresponding to eigenvalue lambda can be found by calculating the null-space of 'A' removing λ multiplied by the In identity matrix. 3. To find the transformation matrix, all we need is to get the real values of a,b,c,d,e, and f. We need to insert our data into our Tx vector(Tx=a*x+b*y+c), and Ty vector(Ty = d*x + e*y + f). If we solve these linear equations we get can get the result. 4. A cofactor is used in finding the inverse, and singular value status of a matrix. It represents the square made from crossing out the row and column of a given cell.