MATH 304Linear AlgebraLecture 7:Inverse matrix (continued).Diagonal matricesDefinition. A square matrix is called diagonal if allnon-diagonal entries are zeros.Example.7 0 00 1 00 0 2, denoted diag(7, 1, 2).Theorem Let A = diag(s1, s2, . . . , sn),B = diag(t1, t2, . . . , tn).Then A + B = diag(s1+ t1, s2+ t2, . . . , sn+ tn),rA = diag(rs1, rs2, . . . , rsn).AB = diag(s1t1, s2t2, . . . , sntn).Identity matrixDefinition. The identity matrix (or unit matrix) isa diagonal matrix with all diagonal entries equal to 1.I1= (1), I2=1 00 1, I3=1 0 00 1 00 0 1.In general, I =1 0 . . . 00 1 . . . 0............0 0 . . . 1.Theorem. Let A be an arbitrary m×n matrix.Then ImA = AIn= A.Inverse matrixDefinition. Let A be an n×n matrix. The inverseof A is an n×n matrix, denoted A−1, such thatAA−1= A−1A = I .If A−1exists then the matrix A is called invertible.Otherwise A is called singular.Let A and B be n×n matrices. If A is invertiblethen we can divide B by A:left division: A−1B, right division: BA−1.Basic properties of inverse matrices:• The inverse matrix (if it exists) is unique.• If A is invertible, so is A−1, and (A−1)−1= A.• If n×n matrices A and B are invertible, so isAB, and (AB)−1= B−1A−1.• If n×n matrices A1, A2, . . . , Akare invertible, sois A1A2. . . Ak, and (A1A2. . . Ak)−1= A−1k. . . A−12A−11.Inverting diagonal matricesTheorem A diagonal matrix D = diag(d1, . . . , dn)is invertible if and only if all diagonal entries arenonzero: di6= 0 for 1 ≤ i ≤ n.If D is invertible then D−1= diag(d−11, . . . , d−1n).d10 . . . 00 d2. . . 0............0 0 . . . dn−1=d−110 . . . 00 d−12. . . 0............0 0 . . . d−1nInverting diagonal matricesTheorem A diagonal matrix D = diag(d1, . . . , dn)is invertible if and only if all diagonal entries arenonzero: di6= 0 for 1 ≤ i ≤ n.If D is invertible then D−1= diag(d−11, . . . , d−1n).Proof: If all di6= 0 then, clearly,diag(d1, . . . , dn) diag(d−11, . . . , d−1n) = diag(1, . . . , 1) = I ,diag(d−11, . . . , d−1n) diag(d1, . . . , dn) = diag(1, . . . , 1) = I .Now suppose that di= 0 for some i. Then for anyn×n matrix B the ith row of the matrix DB is azero row. Hence DB 6= I .Inverting 2-by-2 matricesDefinition. The determinant of a 2×2 matrixA =a bc dis det A = ad − bc.Theorem A matrix A =a bc dis invertible ifand only if det A 6= 0.If det A 6= 0 thena bc d−1=1ad − bcd −b−c a.Theorem A matrix A =a bc dis invertible ifand only if det A 6= 0. If det A 6= 0 thena bc d−1=1ad − bcd −b−c a.Proof: Let B =d −b−c a. ThenAB = BA =ad−bc 00 ad−bc= (ad − bc)I2.In the case det A 6= 0, we have A−1= (det A)−1B.In the case det A = 0, the matrix A is not invertible asotherwise AB = O =⇒ A−1AB = A−1O =⇒ B = O=⇒ A = O, but the zero matrix is singular.Problem. Solve a system4x + 3y = 5,3x + 2y = −1.This system is equivalent to a matrix equation4 33 2xy=5−1.Let A =4 33 2. We have det A = − 1 6= 0.Hence A is invertible. Let’s multiply both sides of the matrixequation by A−1from the left:4 33 2−14 33 2xy=4 33 2−15−1,xy=4 33 2−15−1=1−12 −3−3 45−1=−1319.System of n linear equations in n variables:a11x1+ a12x2+ · · · + a1nxn= b1a21x1+ a22x2+ · · · + a2nxn= b2· · · · · · · · ·an1x1+ an2x2+ · · · + annxn= bn⇐⇒ Ax = b,whereA =a11a12. . . a1na21a22. . . a2n............an1an2. . . ann, x =x1x2...xn, b =b1b2...bn.Theorem If the matrix A is invertible then thesystem has a unique solution, which is x = A−1b.Problem. Solve the matrix equation XA + B = X ,where A =4 −21 1, B =5 23 0.Since B is a 2×2 matrix, it follows that XA and Xare also 2×2 matrices.XA + B = X ⇐⇒ X − XA = B⇐⇒ X (I − A) = B ⇐⇒ X = B(I − A)−1provided that I −A is an invertible matrix.I −A =−3 2−1 0,• I −A =−3 2−1 0,• det(I −A) = (−3) · 0 − 2 · (−1) = 2,• (I −A)−1=120 −21 −3,• X = B(I −A)−1=5 23 0120 −21 −3=125 23 00 −21 −3=122 −160 −6=1 −80 −3.Fundamental results on inverse matricesTheorem 1 Given a square matrix A, the following areequivalent:(i) A is invertible;(ii) x = 0 is the only solution of the matrix equation Ax = 0;(iii) the row echelon form of A has no zero rows;(iv) the reduced row echelon form of A is the identity matrix.Theorem 2 Suppose that a sequence of elementary rowoperations converts a matrix A into the identity matrix.Then the same sequence of operations converts the identitymatrix into the inverse matrix A−1.Theorem 3 For any n×n matrices A and B,BA = I ⇐⇒ AB = I .Row echelon form of a square matrix:∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗invertible case noninvertible
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