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Determinants and Inverses

Determinants and Inverses. Consider the weight suspended by wires problem:. One (poor) way is to find inverse or ‘reciprocal’ of A , A -1 . It is defined that A -1 * A = A * A -1 = I the identity matrix If we know A -1 , then by pre-multiplying the equation:

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Determinants and Inverses

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  1. Determinants and Inverses • Consider the weight suspended by wires problem: One (poor) way is to find inverse or ‘reciprocal’ of A, A-1. It is defined that A-1*A = A*A-1 = I the identity matrix If we know A-1, then by pre-multiplying the equation: A-1 * A * T = A-1 * Y thus T = A-1 * Y How to find the inverse - if one exists (just as 1/0 not exists) ? We shall consider square matrices and one method of finding the inverse - using determinants and cofactor matrices. Note: determinants are not just used to find inverses. In fact, determinants are more useful than inverses. EG1C2 Engineering Maths: Matrix Algebra 3

  2. Determinants determinant of A = det(A) = |A| |a11| = a11 EG1C2 Engineering Maths: Matrix Algebra 3

  3. Cofactors for finding Determinants Det of matrix without ith row and jth column is defined as Mij. e.g for 3*3 matrix The cofactor of element i,j, Cij, is (-1)i+jMij (-1)i+j = 1 or -1 Then the determinant of any n*n matrix is defined as a11 * C11 + a12 * C12 + ... + a1n * C1n Determinant of a 2*2 matrix is: a11 * C11 + a12 * C12 = a11 * (-1)1+1a22 + a12 * (-1)1+2a21= a11 * a22 - a12 * a21 Determinant of a 3*3 matrix: a11 * C11 + a12 * C12 + a13 * C13 EG1C2 Engineering Maths: Matrix Algebra 3

  4. Adjoint Matrix of a square matrix A,AdjA If Cij is the cofactor of aij, then Adj A, = [Cji] = [Cij]T. then the matrix of cofactors of A is: i.e. the transpose of the above EG1C2 Engineering Maths: Matrix Algebra 3

  5. Matrix Inverse (poor method) EG1C2 Engineering Maths: Matrix Algebra 3

  6. Example The cofactors are: EG1C2 Engineering Maths: Matrix Algebra 3

  7. Properties (A-1)-1 = A (AT)-1 = (A-1)T (A * B)-1 = B-1 * A-1 If AT = A-1 then matrix A is an orthogonal matrix. EG1C2 Engineering Maths: Matrix Algebra 3

  8. Applying Inverses to Example Systems Suspended Mass i.e. T1 = 300N and T2 = 360N Check T1*0.96 - T2*0.8 = 288 - 288 = 0 T1*0.28 + T2*0.6 = 84 + 216 = 300  Good! EG1C2 Engineering Maths: Matrix Algebra 3

  9. Electronic Circuit |A| = 18 * (-10-15) - 10 * (0 --15) + 0 * (0-1) = -600 EG1C2 Engineering Maths: Matrix Algebra 3

  10. Thus, i1 = 0.5A, Check: 18 * i1 + 10 * i2 = 9 + 3 = 12 i2 = 0.3A -10 * i2 + 15 * i3 = -3 + 3 = 0 and i3 = 0.2A -i1 + i2 + i3 = -0.5 + 0.3 + 0.2 = 0 Exercise: Use matrix inversion to solve EG1C2 Engineering Maths: Matrix Algebra 3

  11. Stochastic Matrix: what was situation in 1990? By post-multiplying both sides by inverse of transition matrix EG1C2 Engineering Maths: Matrix Algebra 3

  12. Note, 1/0.3 is a scalar, and a k b = k a b EG1C2 Engineering Maths: Matrix Algebra 3

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