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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition D 13 Copyright © Orchard Publications Minors and Cofactors If we remove the elements of its row, and column, the remaining square matrix is called the minor of , and it is denoted as . The signed minor is called the cofactor of and it is denoted as . Example D.6 Matrix is defined as (D.18) Compute the minors , , and the cofactors , and . Solution: and The remaining minors and cofactors are defined similarly. ith jth n 1 A M ij 1 () + M a α A A a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = M 11 M 12 M 13 α 11 α 12 α 13 M 11 a 22 a 23 a 32 a 33 = M 12 a 21 a 23 a 31 a 33 = M 11 a 21 a 22 a 31 a 32 = α 11 1 11 + M 11 M 11 α 12 1 12 + M 12 M 12 α 13 M 13 1 13 + M 13 == M 21 M 22 M 23 M 31 M 32 M 33 ,,,,, α 21 α 22 α 23 α 31 α 32 and α 33

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Appendix D Matrices and Determinants D 14 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Example D.7 Compute the cofactors of matrix defined as (D.19) Solution: (D.20) (D.21) (D.22) (D.23) (D.24) It is useful to remember that the signs of the cofactors follow the pattern below that is, the cofactors on the diagonals have the same sign as their minors. Let be a square matrix of any size; the value of the determinant of is the sum of the products obtained by multiplying each element of any row or any column by its cofactor. A A 12 3 24 2 1 26 = α 11 1 () 11 + 4 2 20 == α 12 1 + 22 1 6 10 α 13 1 13 + 1 2 0 α 21 1 21 + 23 6 α 22 1 + 1 6 9 α 23 1 + 1 2 4 α 31 1 31 + 4 2 8 α 32 1 32 + 8 , α 33 1 33 + 8 + + + + + + + + + + + + + AA
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition D 15 Copyright © Orchard Publications Minors and Cofactors Example D.8 Matrix is defined as (D.25) Compute the determinant of using the elements of the first row. Solution: Check with MATLAB: A=[1 2 3; 2 4 2; 1 2 6]; det(A) % Define matrix A and compute detA ans = 40 We must use the above procedure to find the determinant of a matrix of order or higher.

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