Lectures on Linear Algebra for IT

Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic

11. Determinants 11.1 Inductive Definition of a Determinant 11.2 Determinant and Antisymmetric Forms 11.3 Determinant Evaluation 11.4 Determinant of the Product of Matrices 11.5 Cofactor Expansion Across any Row 11.6 Adjugate Matrix and Matrix Inverse 11.7 Determinant of a Matrix Transpose 11.8 Determinant as a Function of Columns 11.9 Cramer s Rule for Linear Systems

11.1 Inductive Definition of a Determinant We solve the linear system a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2. By performing replacement elementary row operations we obtain [ ] [ a11 a 12 b 1 a 21 a 22 b 2 a 22 r 1 a 12 r 2 ] a 11 a 12 b 1 a 11 a 22 a 12 a 21 0 b 1 a 22 a 12 b 2 [ ] [ ] a11 a 12 b 1 a11 a 12 b 1 a 21 a 22 b 2 a 11 r 2 a 21 r 1 0 a 11 a 22 a 12 a 21 a 11 b 2 b 1 a 21 from where, for a 11 a 22 a 12 a 21 0, is x 1 = b 1a 22 a 12 b 2 a 11 a 22 a 12 a 21, x 2 = a 11b 2 b 1 a 21 a 11 a 22 a 12 a 21.

11.1 Inductive Definition of a Determinant Notice, it is possible to express the nominator and the denominator too by the following matrix function: [ ] a b det = c d a b c d = ad bc. Then, the solution of the system can be writen in the form b 1 a 12 b 2 a 22 a 11 b 1 a 21 b 2 x 1 =, x 2 =, d d where d = a 11 a 22 a 12 a 21 = a 11 a 12 a 21 a 22 Simmilar formulas will be given in general for the linear systems of n equations with n variables.

11.1 Inductive Definition of a Determinant Definition 1 If A = [a ij ] is a square matrix, then the minor of the entry in the i-th row and j-th column is the submatrix of A formed by deleting the i-th row and j-th column, denoted by M A ij. Example 1 A = 1 2 3 4 5 6 7 8 9, M A 12 = [ 4 6 7 9 ].

11.1 Inductive Definition of a Determinant Definition 2 For a real or a complex square matrix A = [a ij ] of the size n the Determinant of A is the number denoted by det A or A which is obtained as follows: D1 For n = 1, is det A = det [a 11 ] = a 11. D2 For n > 1, with assumption that we know how to evaluate determinant of any square matrix of size n 1, is det A = a 11 M A 11 a12 M A 12 + a13 M A 13 + ( 1) n+1 a 1n M A 1n Note: The determinant of an n n matrix is the function of the matrix entries, defined explicitly for n = 1 and defined recursively for n > 1. In general, an n n determinant is defined by determinants of (n 1) (n 1) submatrices.

11.1 Inductive Definition of a Determinant Example 2 1 2 3 1 1 2 1 2 1 = 1 1 2 2 1 2 1 2 1 1 + 3 1 1 1 2 = 1 ( 1 1 2 2 ) 2 (1 1 2 1 ) + 3 (1 2 ( 1) 1 ) = 1 ( 5) 2 3 + 3 1 = 8. Example 3 l 11 0... 0 l 22 0... 0 l 21 l 22... 0 l 32 l 33... 0..... = l 11....... l n1 l n2... l nn l n2 l n3... l nn The above calculation implies: det I = 1. = = l 11 l nn.

11.1 Inductive Definition of a Determinant Definition 3 Given A = [a ij ] we define the (i, j)-cofactor of A to be the number A ij given by A ij = ( 1) i+j M A ij. Then the formula D2 becomes det A = a 11 A 11 + + a 1n A 1n and is called cofactor expansion across the first row of A. Note: What is the number of arithmetic operations to evaluate a n n determinant? Using the formula D2 - sum of n products of a number with (n 1) (n 1) determinant. Using the formula D2 again - sum of n(n 1) products of a number with (n 2) (n 2) determinant. By repeating this procedure we obtain, that the evaluation of a n n determinant requires sum of n! terms, that are products of n numbers. Altogether it is (n 1)n! products. More efficient ways to evaluate a determinant are known.

11.2 Determinant and Antisymmetric Forms Lemma 1 Suppose A = [a ij ] and B = [b ij ] are square matrices whose entries may differ only in the first row. Let α be any scalar, then αra 1 = α det A, r A 1 + r B 1 = det A + det B. Proof: αa 11... αa 1n αra 1 = a 21... a 2n....... = αa 11 A 11 + + αa 1n A 1n = α det A a n1... a nn a 11 + b 11... a 1n + b 1n ra 1 + r B 1 = a 21... a 2n....... = (a 11 + b 11 )A 11 + + (a 1n + b 1n )A 1n a n1... a nn = (a 11 A 11 + + a 1n A 1n ) + (b 11 A 11 + + b 1n A 1n ) = det A + det B.

11.2 Determinant and Antisymmetric Forms Lemma 2 If A=[a ij ] is a square matrix of the size n 2, then det A = r A 1 r A 2 = r A 2 r A 1. Proof: For a 2 2 determinant is ra 1 r A = a 11 a 12 = 2 a 21 a 22 = a 11 a 22 a 12 a 21 = (a 21 a 12 a 22 a 11 ) = ra 2 r A 1. Proof of the general case relies on Cofactor expansion (D2 formula) with further work on determinants of minors. It is more complicated and to avoid lengthy expressions we omit the proof.

11.2 Determinant and Antisymmetric Forms Theorem 1 Let A = [a ij ] be a square matrix of size n 2 and let B = [b ij ] be the matrix obtained from A by interchange of i-th and j-th row. Then r A i det A = r A j i j r A j = r A i i j = det B. Proof: Proof relies on Lemma 2 and is done by mathematical induction (find in literature).

11.2 Determinant and Antisymmetric Forms Theorem 2 Let A and B be square matrices of size n which have the same entries except of the k-the row. Then for any scalar α is αr A k = α det A, r A k + rb k = det A + det B. Proof: Proof relies on Lemma 1 and Theorem 1 and again is done by mathematical induction (find in literature). Note: The meaning of preceding two Theorems (1 and 2) can be summarized in one sentence: A determinant is an antisymmetric bilinear form of any two rows of a matrix.

11.2 Determinant and Antisymmetric Forms Consequent properties of determinants: For any square matrix A the following holds: 1. If A has two rows equal, then det A = 0. 2. If A has a zero row, then det A = 0. 3. If another square matrix B is the same as A except of the k-th row and r B k = r A k + αr A l, k l, then det B = det A. 4. If the rows of A are linearly dependent, then det A = 0.

11.3 Determinant Evaluation Here we describe in words how simply elementary row operations affect the value of a determinant. 1. An interchange of two rows alternates the sign of the determinant. 2. If a row is multiplied by a scalar α, also the value of the determinant is multiplied by α. 3. Performing replacement operation (add a multiple of one row to another row) does not change the value of the determinant. So the elementary row operations are the powerfull tool for evaluating determinants, if they are used to transform a matrix into a form convenient for the determinant evaluation. So far for us the convenient form is a lower triangular form, then the determinant value is the product of diagonal entries (as we have seen in Example 3).

Example 4 3 2 1 2 0 2 3 1 1 = 2 6 0 0 4 1 0 3 1 1 11.3 Determinant Evaluation 6 3 0 8 2 0 3 1 1 = 2 = 2 ( 6) 1 ( 1) = 12 +r 3 +2r 3 = 6 3 0 4 1 0 3 1 1 3r 2 = Example 5 0 1 1 1 0 1 1 1 0 r 3 r 2 = = 0 1 1 1 1 0 1 0 1 2 0 0 1 1 0 1 0 1 r 3 1 1 0 = 1 1 0 1 0 1 = ( 2) 1 1 = 2 r 2 =

11.4 Determinant of the Product of Matrices Theorem 3 If A and B are square matrices of the size n, then det(ab) = det A det B. Proof: First suppose A is diagonal. Then d 1 0... 0 0 d 2... 0 A =.......... 0 0... d n Then det(ab) = d 1 r B 1. d n r B n = d 1 d n det B = det A det B.

11.4 Determinant of the Product of Matrices Proof: (kont.) For any matrix A exists a sequence of elementary transformation matrices T 1,..., T k, that consists of l interchanges and k l replacements, so that d 1 0... 0 0 d 2... 0 T k T 1 A =...... = D. 0 0... d n Because left multiplication by an elementary transformation matrix T is equivalent to performing the elementary row operation on the multiplied matrix, we can write det(ab) = ( 1) l det(t k T 1 AB) = ( 1) l det(db) = = ( 1) l det D det B = ( 1) l det(t k T 1 A) det B = = ( 1) l ( 1) l det A det B = det A det B.

11.5 Cofactor Expansion Across any Row Theorem 4 Let A = [a ij ] be a square matrix of the size n > 1. Then for any row index k is det A = a k1 A k1 + + a kn A kn. Corollary 1 Let A = [a ij ] be a square matrix of the size n > 1. 1. If k, l are the two different row indexes of rows of A, then a k1 A l1 + + a kn A ln = 0. 2. If A is a triangular matrix, then det A = a 11 a nn.

11.6 Adjugate Matrix and Matrix Inverse Definition 4 Let A be any square matrix of the size n > 1. Then adjugate matrix of A denoted by à is the square matrix of the same size as A defined by A 11... A n1 à =....., A 1n... A nn where A ij = ( 1) i+j M A ij. (The ij-th cofactor of A.) Example 6 For A = 3 2 1 2 0 2 3 1 1 is à = 2 3 4 8 6 4 2 3 4.

Example 6 (cont.) 11.6 Adjugate Matrix and Matrix Inverse And here is calculation of the cofactors individually. For A = 3 2 1 2 0 2 is à = 2 3 4 8 6 4. 3 1 1 2 3 4 A 11 = 0 2 1 1 = 2, A 12 = 2 2 3 1 = ( 2 6) = 8, A 13 = 2 0 3 1 = 2, A 21 = 2 1 1 1 = ( 2 1) = 3, A 22 = 3 1 3 1 = 3 3 = 6, A 23 = 3 2 3 1 = 3, A 31 = 2 1 0 2 = 4, A 32 = 3 1 2 2 = 4, A 33 = 3 2 2 0 = 4.

11.6 Adjugate Matrix and Matrix Inverse Theorem 5 Let A be a square matrix of size n > 1. Then for det A 0 the inverse of A is A 1 = 1 det AÃ. Example 7 For the matrix A from Example 6 the inverse matrix is 3 2 1 1 2 0 2 = 1 2 3 4 8 6 4 = 1 6 2 3 1 1 12 2 3 4 1 4 1 3 3 1 2 1 3 1 6 1 4 1 3. Corollary 2 A matrix A is invertible (regular) if and only if det A 0. Proof: For A a singular matrix, the rows of A are dependent. Therefore, by the Corollary 4, det A = 0. The opposite direction of the claim is true by Theorem 5.

11.7 Determinant of a Matrix Transpose Theorem 6 Let A be a square matrix. Then det A = det A. Proof: Every permutation matrix P can be written as a product of elementary permutation matrices P = P 1 P k, where each P i is symmetric. Then P = P k P 1 and det P = P k P 1 = det P. This is true also for any triangular matrix, since the determinant is then equal to the product of diagonal entries and transpose of a triangular matrix is again triangular with the same diagonal. For A is any square matrix, then according to LUP factorization there exist a lower triangular matrix L, an upper triangular matrix U, and a permutation matrix P such, that A = LUP. By the Theorem 3, about the determinant of product of matrices is det A = det(p U L ) = P U L = L U P = det(lup) = det A.

11.8 Determinant as a Function of Columns The Theorem 6 implies, that a determinant as well as a function of rows can be considered as a function of columns with all it s properties. For instance a determinant is an antisymmetric bilinear form of any two matrix columns. Theorem 7 Let A be any square matrix of the size n > 1. Then for i = 1,..., n is det A = a 1i A 1i + + a ni A ni. Note: A determinant can be evaluated by the cofactor expansion down any column of a matrix.

11.9 Cramer s Rule for Linear Systems We might use the matrix inverse of an invertible matrix A of size n > 1 to solve the linear system Ax = b. Then the solution vector x has the entries x i, where x i = [ A 1 b ] i = 1 det A (A 1ib 1 + + A ni b n ). The i-th column minors do not contain entries of the i-th column of A. Therefore the above expression in the parentheses is the same as cofactor expansion down the i-th column of the matrix i i A b i = [ b ] = [ ] s A 1... s A i 1 b s A i+1... s A n, which is obtained from A by replacing the i-th column with the right side vector b. And so we can write x i = det Ab i det A, i = 1,..., n The above formula is called Cramer s rule.

Example 8 11.10 Cramer s Rule for Linear Systems Use Cramer s rule to solve the system 3x 1 + 2x 2 + x 3 = 6 2x 1 + 2x 3 = 0 3x 1 x 2 x 3 = 0 Solution: Individually we evaluate all the determinants needed to apply Cramer s rule. 3 2 1 A = 2 0 2 3 1 1 = 20 and A b 1 = 6 2 1 0 0 2 0 1 1 By Cramer s rule, = 12, Ab 2 = 3 6 1 2 0 2 3 0 1 = 48, Ab 3 = 3 2 6 2 0 0 3 1 0 x 1 = Ab 1 det A = 3 5, x 2 = Ab 2 det A = 12 5, x 3 = Ab 3 det A = 3 5. = 12