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ENE 206 - MATHLAB ®

ENE 206 - MATHLAB ®. Lecture 3: Matrix Review. Determinant. To use MATLAB to compute determinants: Enter the determinant as an array. Use ‘ det ’ command to evaluate the determinant. >> A = [1 2; 3 4] A = 1 2 3 4 >> det(A) ans = -2. Inverse matrix.

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ENE 206 - MATHLAB ®

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  1. ENE 206 - MATHLAB® Lecture 3: Matrix Review

  2. Determinant • To use MATLAB to compute determinants: • Enter the determinant as an array. • Use ‘det’ command to evaluate the determinant. • >> A = [1 2; 3 4] • A = • 1 2 • 3 4 • >> det(A) • ans = • -2

  3. Inverse matrix • To compute the inverse of the matrix, the command ‘inv’ is used. • For example, we have two equations: 2x + 9y=5 and 3x - 4y = 7. Solve these equations using the matrix inverse method. (x = A-1b) A = [2 9; 3 -4]; b = [5;7]; x = inv(A)*b x = 2.3714 0.0286

  4. The Left-Division Method • The inverse matrix method works only if the matrix A is square(the number of unknowns equals the number of equation). • Even if A is square, the method does not work for singular matrix (|A| = 0). • MATLAB provides another method to solve the equation Ax = b based on ‘Gauss elimination’ called ‘the left-division method’. • The LDM uses fewer internal multiplications and divisions; therefore, it is faster and more accurate than the matrix inverse.

  5. The Left-Division Method • To use the LDM to solve for x, type ‘x = A\b’. • This works for some cases where the number of unknowns does not equal the number of equations. • Anyway, simply check first to see if |A| 0 before using any of these methods. • If |A|= 0 or if the number of equations does not equal the number of unknowns, it will be in a case of ‘underdetermined systems.’

  6. Underdetermined system • This system does not contain enough information to solve for all unknown variables. • The matrix inverse method and Cramer’s method will not work. • When there are more equations than unknowns, the LDM will give a solution with some of the unknowns being set equal to zero.

  7. Underdetermined system • For example, x + 3y = 6. Actually, an infinite number of solutions satisfy this equation. • By using the LDM method, the solutions will be: A = [1 3]; b = 6; x = A\b x = 0 (x = 0) 2 (y = 2)

  8. Matrix Rank • An m x n matrix A has a rank r  1 if and only if |A|contains a nonzero r x r determinant and every square sub-determinant with r + 1 or more is zero. • For example, A = [3 -4 1; 6 10 2; 9 -7 3]; D =det (A) R = rank (A) D = 0 R = 2

  9. Unique solutions • The set Ax = b with m equations and n unknowns has solutions if and only if rank[A] = rank[A b]. • Let r = rank[A] and rank[A] = rank[A b]: • If r = n, then the solution is unique. • If r < n, an infinite number of solutions exists and r unknown variables can be expressed as linear combinations of the other n – r unknown variables, whose values are arbitrary.

  10. Example of unique solution • Determine whether the following set has a unique solution, and if so, find it: • 3x – 2y + 8z = 48 • -6x + 5y + z = -12 • 9x + 4y +2z = 24

  11. Example of unique solution A = [3 -2 8; -6 5 1; 9 4 2]; b = [48;-12;24]; R = rank (A) Rab = rank ([A b]) x = A\b R = 3 Rab = 3 x = 2.0000 -1.0000 5.0000

  12. To be continued….

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