Lect1-06-web

# Lect1-06-web - MATH 304 Fall 2011 Linear Algebra Help...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Help sessions: The Math 304/309/311/323 Help Session will be held on MW from 7:30-10:00pm and on TR from 6:30-9:00 p.m., BLOC 160 Homework assignment #3 (due Thursday, September 22) All problems are from Leon’s book (8th edition). Section 2.1: 3b, 3c, 3d, 3e, 3f, 3g Section 2.2: 3f, 4, 7c, 7d MATH 304 Linear Algebra Lecture 6: Transpose of a matrix. Determinants. Transpose of a matrix Deﬁnition. Given a matrix A, the transpose of A, denoted AT , is the matrix whose rows are columns of A (and whose columns are rows of A). That is, if A = (aij ) then AT = (bij ), where bij = aji . 14 T 123 Examples. = 2 5 , 456 36 T 7 T 47 47 8 = (7, 8, 9), = . 70 70 9 Properties of transposes: • (AT )T = A • (A + B )T = AT + B T • (rA)T = rAT • (AB )T = B T AT • (A1 A2 . . . Ak )T = AT . . . AT AT 21 k • (A−1)T = (AT )−1 Deﬁnition. A square matrix A is said to be symmetric if AT = A. For example, any diagonal matrix is symmetric. Proposition For any square matrix A the matrices B = AAT and C = A + AT are symmetric. Proof: B T = (AAT )T = (AT )T AT = AAT = B , C T = (A + AT )T = AT + (AT )T = AT + A = C . Determinants Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (aij )1≤i ,j ≤n is denoted det A or a11 a12 a21 a22 . . . . . . an1 an2 . . . a1n . . . a2n ... . . . . . . . ann Principal property: det A = 0 if and only if a system of linear equations with the coeﬃcient matrix A has a unique solution. Equivalently, det A = 0 if and only if the matrix A is invertible. Deﬁnition in low dimensions Deﬁnition. det (a) = a, ab = ad − bc , cd a11 a12 a13 a21 a22 a23 = a11a22 a33 + a12a23a31 + a13a21a32 − a31 a32 a33 −a13a22a31 − a12 a21a33 − a11a23 a32. +: −: * ∗ ∗ ∗ ∗ * ∗ * ∗ ∗ * ∗ ∗ ∗ ∗ , ∗ * * * ∗ ∗ , * ∗ ∗ * ∗ ∗ * ∗ ∗ ∗ * , ∗ ∗ ∗ , * ∗ * ∗ ∗ ∗ * * ∗ ∗ ∗ ∗ * * ∗ . ∗ ∗ * . ∗ Examples: 2×2 matrices 10 = 1, 01 −2 5 = − 6, 03 30 = − 12, 0 −4 70 = 14, 52 0 −1 = 1, 10 00 = 0, 41 −1 3 = 0, −1 3 21 = 0. 84 Examples: 3×3 matrices 3 −2 0 1 0 1 = 3 · 0 · 0 + (−2) · 1 · (−2) + 0 · 1 · 3 − −2 3 0 − 0 · 0 · (−2) − (−2) · 1 · 0 − 3 · 1 · 3 = 4 − 9 = −5, 146 0 2 5 =1·2·3+4·5·0+6·0·0− 003 − 6 · 2 · 0 − 4 · 0 · 3 − 1 · 5 · 0 = 1 · 2 · 3 = 6. General deﬁnition The general deﬁnition of the determinant is quite complicated as there is no simple explicit formula. There are several approaches to deﬁning determinants. Approach 1 (original): an explicit (but very complicated) formula. Approach 2 (axiomatic): we formulate properties that the determinant should have. Approach 3 (inductive): the determinant of an n×n matrix is deﬁned in terms of determinants of certain (n − 1)×(n − 1) matrices. Mn (R): the set of n×n matrices with real entries. Theorem There exists a unique function det : Mn (R) → R (called the determinant) with the following properties: • if a row of a matrix is multiplied by a scalar r , the determinant is also multiplied by r ; • if we add a row of a matrix multiplied by a scalar to another row, the determinant remains the same; • if we interchange two rows of a matrix, the determinant changes its sign; • det I = 1. Corollary 1 Suppose A is a square matrix and B is obtained from A applying elementary row operations. Then det A = 0 if and only if det B = 0. Corollary 2 det B = 0 whenever the matrix B has a zero row. Hint: Multiply the zero row by the zero scalar. Corollary 3 det A = 0 if and only if the matrix A is not invertible. Idea of the proof: Let B be the reduced row echelon form of A. If A is invertible then B = I ; otherwise B has a zero row. Row echelon form of a square matrix A: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ det A = 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ det A = 0 3 −2 0 Example. A = 1 0 1, det A =? −2 3 0 In the previous lecture we have transformed the matrix A into the identity matrix using elementary row operations: • • • • • • • • interchange the 1st row with the 2nd row, add −3 times the 1st row to the 2nd row, add 2 times the 1st row to the 3rd row, multiply the 2nd row by −0.5, add −3 times the 2nd row to the 3rd row, multiply the 3rd row by −0.4, add −1.5 times the 3rd row to the 2nd row, add −1 times the 3rd row to the 1st row. 3 −2 0 Example. A = 1 0 1, det A =? −2 3 0 In the previous lecture we have transformed the matrix A into the identity matrix using elementary row operations. These included two row multiplications, by −0.5 and by −0.4, and one row exchange. It follows that det I = − (−0.5) (−0.4) det A = (−0.2) det A. Hence det A = −5 det I = −5. Other properties of determinants • If a matrix A has two det A = 0. a1 a2 b1 b2 a1 a2 identical rows then a3 b3 = 0 a3 • If a matrix A has two rows proportional then det A = 0. a1 a2 a3 a1 a2 a3 b1 b2 b3 = r b1 b2 b3 = 0 a1 a2 a3 ra1 ra2 ra3 Distributive law for rows • Suppose that matrices X , Y , Z are identical except for the i th row and the i th row of Z is the sum of the i th rows of X and Y . Then det Z = det X + det Y . ′ ′ ′ ′ ′ ′ a1 +a1 a2+a2 a3 +a3 a1 a2 a3 a1 a2 a3 b1 b2 b3 = b1 b2 b3 + b1 b2 b3 c1 c2 c3 c1 c2 c3 c1 c2 c3 • Adding a scalar multiple of one row to another row does not change the determinant of a matrix. a1 + rb1 a2 + rb2 a3 + rb3 = b1 b2 b3 c1 c2 c3 a1 a2 a3 rb1 rb2 rb3 a1 a2 a3 = b1 b2 b3 + b1 b2 b3 = b1 b2 b3 c1 c2 c3 c1 c2 c3 c1 c2 c3 Deﬁnition. A square matrix A = (aij ) is called upper triangular if all entries below the main diagonal are zeros: aij = 0 whenever i > j . • The determinant of an upper triangular matrix is equal to the product of its diagonal entries. a11 a12 a13 0 a22 a23 = a11a22a33 0 0 a33 • If A = diag(d1, d2, . . . , dn ) then det A = d1 d2 . . . dn . In particular, det I = 1. Determinant of the transpose • If A is a square matrix then det AT = det A. a1 b1 c1 a1 a2 a3 a2 b2 c2 = b1 b2 b3 a3 b3 c3 c1 c2 c3 Columns vs. rows • If one column of a matrix is multiplied by a scalar, the determinant is multiplied by the same scalar. • Interchanging two columns of a matrix changes the sign of its determinant. • If a matrix A has two columns proportional then det A = 0. • Adding a scalar multiple of one column to another does not change the determinant of a matrix. Submatrices Deﬁnition. Given a matrix A, a k ×k submatrix of A is a matrix obtained by specifying k columns and k rows of A and deleting the other columns and rows. 1234 ∗2∗4 10 20 30 40 → ∗ ∗ ∗ ∗ → 2 4 59 3579 ∗5∗9 Given an n×n matrix A, let Mij denote the (n − 1)×(n − 1) submatrix obtained by deleting the i th row and the j th column of A. 3 −2 0 Example. A = 1 0 1. −2 3 0 M11 = M21 = M31 = 01 , M12 = 30 11 , M13 = −2 0 10 , −2 3 −2 0 , M22 = 30 30 , M23 = −2 0 3 −2 , −2 3 −2 0 , M32 = 01 30 , M33 = 11 3 −2 . 10 Row and column expansions Given an n×n matrix A = (aij ), let Mij denote the (n − 1)×(n − 1) submatrix obtained by deleting the i th row and the j th column of A. Theorem For any 1 ≤ k , m ≤ n we have that n (−1)k +j akj det Mkj , det A = j =1 (expansion by kth row ) n (−1)i +maim det Mim . det A = i =1 (expansion by mth column) Signs for row/column expansions + − + − . . . − + − + . . . + − + − . . . − + − + . . . ··· · · · · · · · · · ... 123 Example. A = 4 5 6. 789 Expansion by the 1st row: ∗2∗ ∗∗ 3 1 ∗∗ ∗ 5 6 4 ∗ 6 4 5 ∗ ∗ 89 7∗9 78 ∗ det A = 1 56 46 45 −2 +3 89 79 78 = (5 · 9 − 6 · 8) − 2(4 · 9 − 6 · 7) + 3(4 · 8 − 5 · 7) = 0. 123 Example. A = 4 5 6. 789 Expansion by the 2nd 1∗ ∗2∗ 4 ∗ 6 ∗ 5 7∗9 7∗ det A = −2 column: 3 1∗3 ∗ 4 ∗ 6 9 ∗8∗ 46 13 13 +5 −8 79 79 46 = −2(4 · 9 − 6 · 7) + 5(1 · 9 − 3 · 7) − 8(1 · 6 − 3 · 4) = 0. 123 Example. A = 4 5 6. 789 Subtract the 1st row from the 2nd row and from the 3rd row: 123 123 123 4 5 6 = 3 3 3 = 3 3 3 =0 789 789 666 since the last matrix has two proportional rows. ...
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