Determinant of a Matrix
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1 13 March 2018
2 Goals We will define determinant of SQUARE matrices, inductively, using the definition of Minors and cofactors. We will see that determinant of triangular matrices is the product of its diagonal elements. Determinants are useful to compute the inverse of a matrix and solve linear systems of equations (Cramer s rule).
3 Overview of the definition Given a square matrix A, the determinant of A will be defined as a scalar, to be denoted by det(a) or A. We define determinant inductively. That means, we first define determinant of 1 1 and 2 2 matrices. Use this to define determinant of 3 3 matrices. Then, use this to define determinant of 4 4 matrices and so.
4 For a 1 1 matrix A = [a] define det(a) = A = a. Let ( ) a b A = define det(a) = A = ad bc. c d
5 Example 1 Preview Let A = ( ) then det(a) = A = 2 ( 2) 17 3 = 53
6 Example 2 Preview Let A = ( ) then det(a) = A = = 0.
7 Minors of 3 3 matrices Let A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Then, the Minor M ij of a ij is defined to be the determinant of the 2 2 matrix obtained by deleting the i th row and j th column.
8 For example M 22 = a 11 a 13 a 31 a 33 = a 11 a 13 a 31 a 33 Like wise M 11 = a 22 a 23 a 32 a 33, M 23 = a 11 a 12 a 31 a 32, M 32 = a 11 a 13 a 21 a 23.
9 Cofactors of 3 3 matrices Let A the 3 3 matrix as in the above frame. Then, the Cofactor C ij of a ij is defined, by some sign adjustment of the minors, as follows: C ij = ( 1) i+j M ij For example, using the above frame C 11 = ( 1) 1+1 M 11 = M 11 = a 22 a 33 a 23 a 33 C 23 = ( 1) 2+3 M 23 = M 23 = (a 11 a 32 a 12 a 31 ) C 32 = ( 1) 3+2 M 32 = (a 11 a 23 a 13 a 21 ).
10 Let A be the 3 3 matrix as above. Then the determinant of A is defined by det(a) = A = a 11 C 11 + a 12 C 12 + a 13 C 13 This definition may be called definition by expansion by cofactors, along the first row. It is possible to define the same by expansion by second of third row, which we will be discussed later.
11 Example 3 Preview Let A = Compute the minor M 11, M 12, M 13, the cofactors C 11, C 12, C 13 and the determinant of A.
12 Solution: Preview Then minors M 11 = , M 12 = , M 13 = Or M 11 = 2, M 12 = 3, M 13 = 1
13 Continued Preview So, the cofactors C 11 = ( 1) 1+1 M 11 = 2, C 12 = ( 1) 1+2 M 12 = 3, So, C 13 = ( 1) 1+3 M 13 = 1 A = a 11 C 11 +a 12 C 12 +a 13 C 13 = 2 ( 2)+1 ( 3)+1 ( 1) = 8
14 Inductive process of definition We defined determinant of size 3 3, using the determinant of 2 2 matrices. Now we can do the same for 4 4 matrices. This means first define minors, which would be determinant of 3 3 matrices. Then, define Cofactors by adjusting the sign of the Minors.Then, use the cofactors fo define the determiant of the 4 4 matrix. Then, we can define minors, cofactors and determinant of 5 5 matrices. The process continues.
15 We assume that we know how to define determiant of (n 1) (n 1) matrices. Let a 11 a 12 a 13 a 1n a 21 a 22 a 13 a 2n A = a 31 a 32 a 33 a 3n a n1 a n2 a n3 a nn be a square matrix of size n n. The minor M ij of a ij is defined to be the determinant of the (n 1) (n 1) matrix obtained by deleting the i th row and j th column.
16 Cofactors and Detarminant of n n Matrices Let A be a n n matrix. Define C ij = ( 1) i+j M ij which iscalled the cofactor of a ij. Define det(a) = A = n a 1j C 1j = a 11 C 11 +a 12 C a 1n C 1n j=1 This would be called a definiton by expasion by cofactors, along first row.
17 Alternative Method for 3 3 matrices: A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Form a new 3 5 matrix by adding first and second column to A: a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a a 32
18 Continued Then A can be computed as follows: add the product of all three entries in the three left to right diagonals. add the product of all three entries in the three right to left diagonals. Then, A is the difference.
19 Definition. Definitions. Let A be a n n matrix. We say A is Upper Triangular matrix, all entries of A below the main diagonal (left to right) are zero. In notations, if a ij = 0 for all i > j. We say A is Lower Triangular matrix, all entries of A above he main diagonal (left to right) are zero. In notations, if a ij = 0 for all i < j.
20 Theorem Preview Theorem Let A be a triangular matrix of order n. Then A is product of the main-diagonal entries. Notationally, A = a 11 a 22 a nn. Proof. The proof is easy when n = 1, 2. We prove it when n = 3. Let use assume A is lower triangular. So, a A = a 21 a 22 0 a 31 a 32 a 33
21 Continued Preview We expand by the first row: A = a 11 C C C 13 = a 11 C 11 = a 11 ( 1) 1+1 a 22 0 a 32 a 33 = a 11a 22 a 33 For upper triangular matrices, we can prove similarly, by column expansion. For higher order matrices, we can use mathematical induction.
22 Example Compute the determiant, by expansion by cofactors, of A = Solution. The cofactors C 11 = ( 1) = 8, C 12 = ( 1) = 2
23 C 13 = ( 1) So, A = a 11 C 11 + a 12 C 12 + a 13 C 13 = = ( 1) ( 4) = 2
24 Example Let A = Compute det(a). Solution. This is an upper triangular matrix. So, A is the product of the diagonal entries. So A = 3 ( 7) 4 5 = 420.
25 Example Solution. So, Solve x x 1 = 0 (x + 3)(x 1) 1 ( 4) = 0 or x 2 + 2x + 1 = 0 (x + 1) 2 = 0 or x = 1.
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