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### Determinants - Mathematics

• 2. Introduction Every square matrix has associated with it a scalar called its determinant. Given a matrix A, we use det(A) or |A| to designate its determinant. We can also designate the determinant of matrix A by replacing the brackets by vertical straight lines. For example, A=[2 1 0 3 ] det(A)=∣ 2 1 0 3 ∣ Definition 1: The determinant of a 1×1 matrix [a] is the scalar a. Definition 2: The determinant of a 2×2 matrix is the scalar ad-bc. For higher order matrices, we will use a recursive procedure to compute determinants. [a b c d]
• 3. Expansion by Cofactors Definition 1: Given a matrix A, a minor is the determinant of square submatrix of A. Definition 2: Given a matrix A=[aij] , the cofactor of the element aij is a scalar obtained by multiplying together the term (-1)i+j and the minor obtained from A by removing the ith row and the jth column. In other words, the cofactor Cij is given by Cij = (−1)i+j Mij. For example, A= [ a11 a12 a13 a21 a22 a23 a31 a32 a33 ] M21=∣ a12 a13 a32 a33 ∣ M22=∣ a11 a13 a31 a33 ∣ ⇒C21=(−1)2+1 M 21=−M21 ⇒C22=(−1) 2+2 M22=M22
• 4. Expansion by Cofactors To find the determinant of a matrix A of arbitrary order, a)Pick any one row or any one column of the matrix; b)For each element in the row or column chosen, find its cofactor; c)Multiply each element in the row or column chosen by its cofactor and sum the results. This sum is the determinant of the matrix. In other words, the determinant of A is given by det( A)=∣A∣=∑ j=1 n aij Cij=ai1 Ci1+ai2 Ci2+⋯+ain Cin det ( A )=∣A∣=∑ i=1 n aij Cij=a1j C1j+a2j C2j +⋯+anj Cnj ith row expansion jth column expansion
• 5. Example 1: We can compute the determinant 1 2 3 4 5 6 7 8 9 =T by expanding along the first row, ( ) ( ) ( ) 1 1 1 2 1 35 6 4 6 4 5 1 2 3 8 9 7 9 7 8 + + + = × − + × − + × −T 3 12 9 0= − + − = Or expand down the second column: ( ) ( ) ( ) 1 2 2 2 3 24 6 1 3 1 3 2 5 8 7 9 7 9 4 6 + + + = × − + × − + × −T 12 60 48 0= − + = Example 2: (using a row or column with many zeroes) ( ) 2 3 1 5 0 1 5 2 1 1 1 3 1 3 1 0 + = × − − − 16=
• 6. Properties of determinants Property 1: If one row of a matrix consists entirely of zeros, then the determinant is zero. Property 2: If two rows of a matrix are interchanged, the determinant changes sign. Property 3: If two rows of a matrix are identical, the determinant is zero. Property 4: If the matrix B is obtained from the matrix A by multiplying every element in one row of A by the scalar λ, then | B|= λ|A|. Property 5: For an n × n matrix A and any scalar λ, det(λA)= λn det(A).
• 7. Properties of determinants Property 6: If a matrix B is obtained from a matrix A by adding to one row of A, a scalar times another row of A, then |A|=|B|. Property 7: det(A) = det(AT ). Property 8: The determinant of a triangular matrix, either upper or lower, is the product of the elements on the main diagonal. Property 9: If A and B are of the same order, then det(AB)=det(A) det(B).
• 8. Inversion Theorem 1: A square matrix has an inverse if and only if its determinant is not zero. Below we develop a method to calculate the inverse of nonsingular matrices using determinants. Definition 1: The cofactor matrix associated with an n × n matrix A is an n × n matrix Ac obtained from A by replacing each element of A by its cofactor. Definition 2: The adjugate of an n × n matrix A is the transpose of the cofactor matrix of A: Aa = (Ac )T
• 9.  Find the adjugate of Solution: The cofactor matrix of A: A= [−1 3 2 0 −2 1 1 0 −2 ] [ ∣−2 1 0 −2 ∣ −∣0 1 1 −2 ∣ ∣0 −2 1 0 ∣ −∣ 3 2 0 −2 ∣ ∣ −1 2 1 −2 ∣ −∣ −1 3 1 0 ∣ ∣ 3 2 −2 1 ∣ −∣ −1 2 0 1 ∣ ∣ −1 3 0 −2 ∣] = [4 1 2 6 0 3 7 1 2 ] Aa = [4 6 7 1 0 1 2 3 2 ] Example of finding adjugate
• 10. Inversion using determinants Theorem 2: A Aa = Aa A = |A| I . If |A| ≠ 0 then from Theorem 2, A(Aa ∣A∣)=(Aa ∣A∣)A=I A −1 = 1 ∣A∣ A a if ∣A∣≠0 That is, if |A| ≠ 0, then A-1 may be obtained by dividing the adjugate of A by the determinant of A. For example, if then A=[a b c d ], A−1 = 1 ∣A∣ Aa = 1 ad−bc [d −b −c a ]
• 11. Use the adjugate of to find A-1A= [−1 3 2 0 −2 1 1 0 −2 ] Aa = [4 6 7 1 0 1 2 3 2 ] ∣A∣=(−1)(−2)(−2)+(3)(1)(1)−(1)(−2)(2)=3 A −1 = 1 ∣A∣ A a = 1 3 [ 4 6 7 1 0 1 2 3 2 ]= [ 4 3 2 7 3 1 3 0 1 3 2 3 1 2 3 ] Inversion using determinants: example
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