Please enable JavaScript.

Coggle requires JavaScript to display documents.

Quantitative Methods (Linear Equatios (it is generally true that a system…

- - - - if an equation 0 = 0 is produced, it is discarded and the procedure is continued.
      - if an equation 0 = a is produced, where a is a nonzero scalar, the procedure is stopped: in this case, the system has no solution.
      - at each step of the procedure, a new variable is made basic: it has coefficient 1 in one of the equations and 0 in all the others.
      - the procedure stops when each equation has a basic variable associated with it. Say "p"equations remain (remember that some of the original "m: equations mays have been discarded).
        
        when n = p, the system has a unique solution.
        
        when n > p, then p variables are basic and the remaining n - p are nonbasic. Any choice of the nonbasic variable yields a solution of the linear system. therefore the system has infinitely many solutions.
- - - - a matrix is singular if its columns form a set of linearly dependent vectors
        nonsingular if its columns are linearly independent
        if B is the inverse of A, then A is the inverse of B
      - Important property:
        when A is a nonsingular matrix (one that have inverse), this linear system has a unique solution, which cna be obtained by multiplying the inverse matrix on the equation
        Ax=b
        A-1Ax=A-1b
        Ix=A-1b
        the unique solution to the system of linear equations is
        x=a-1*b
      - det(a1)=a11>0 and det(A2)=a11a22-a21a22>0 then is a positive definite
        det(A1)=a11<0 and det(A2)=a11*a22-a21*a12>0 then it is a negative definite
        de(A1)=a11>=0 and det(A2)=a11a22-a21a12>=0 then it is a positive semidefinite
        det(A1)=a11<=0 and det(A2)=a11a22-a21a12>=0 then it is a negative semidefinite
        det(a1)=a11>0 and det(A2)=a11a22-a21a12<0 the it is none of the above
- - - - det of A4x4 uses Laplace Theorem
        If entire line or column is 0, Δ=0
        if 2 lines or columns are equal, Δ=0
        two lines or columns that are multiple, Δ=0
        triangular matrices, Δ=multiply elements of the diagonal
        if you multiply a line or a column of a matrice by an scalar, the determinant is multiplied by the same factor.
        if you multiply an entire matrice by an scalar, the determinate is multiplied by how many columns are multiplied by the scalar, Anxn and det(A)=D then det(kA) = (k^n)D
        Laplace Determinant A4x4
        D=Sum(aijAij)
        D = a12A12 + a22A22 + a32A32 + a42A42
        Aij=(-1)^î+jdetA22, where detA22 is the det of complementary matrice (all the rest)
        a12 = a12
        if you change lines by columns, -1xΔ
        if you sum a multiple of one line into the other line the det does not change
        Binet Theorem: det(AB) = det(A)det(B)
        det(A+B)≠det(A)+det(B)
        det(a-1)? / A-1A=I / det(a-1)*det(A) = det(I) then det(a-1) = 1/det(A)
        de(AT) = det (A)