MATH 304Linear AlgebraLecture 6:Transpose of a matrix.Determinants.Transpose of a matrixDefinition. Given a matrix A, the transpose of A,denoted AT, is the matrix whose rows are columnsof A (and whose columns are rows of A). That is,if A = (aij) then AT= (bij), where bij= aji.Examples.1 2 34 5 6T=1 42 53 6,789T= (7, 8, 9),4 77 0T=4 77 0.Properties of transposes:• (AT)T= A• (A + B)T= AT+ BT• (rA)T= rAT• (AB)T= BTAT• (A1A2. . . Ak)T= ATk. . . AT2AT1• (A−1)T= (AT)−1Definition. A square matrix A is said to besymmetric if AT= A.For example, any diagonal matrix is symmetric.Proposition For any square matrix A the matricesB = AATand C = A + ATare symmetric.Proof:BT= (AAT)T= (AT)TAT= AAT= B,CT= (A + AT)T= AT+ (AT)T= AT+ A = C .DeterminantsDeterminant is a scalar assigned to each square matrix.Notation. The determinant of a matrixA = (aij)1≤i,j≤nis denoted det A ora11a12. . . a1na21a22. . . a2n............an1an2. . . ann.Principal property: det A = 0 if and only if thematrix A is singular.Definition in low dimensionsDefinition. det (a) = a,a bc d= ad − bc,a11a12a13a21a22a23a31a32a33= a11a22a33+ a12a23a31+ a13a21a32−−a13a22a31− a12a21a33− a11a23a32.+ :* ∗ ∗∗* ∗∗ ∗ *,∗* ∗∗ ∗** ∗ ∗,∗ ∗** ∗ ∗∗ * ∗.− :∗ ∗*∗ * ∗* ∗ ∗,∗* ∗* ∗ ∗∗ ∗ *,* ∗ ∗∗ ∗ *∗ * ∗.Examples: 2×2 matrices1 00 1= 1,3 00 −4= − 12,−2 50 3= − 6,7 05 2= 14,0 −11 0= 1,0 04 1= 0,−1 3−1 3= 0,2 18 4= 0.Examples: 3×3 matrices3 −2 01 0 1−2 3 0= 3 · 0 · 0 + (−2) · 1 · (−2) + 0 · 1 · 3 −− 0 · 0 · (−2) − (−2) · 1 · 0 − 3 · 1 · 3 = 4 − 9 = −5,1 4 60 2 50 0 3= 1 · 2 · 3 + 4 · 5 · 0 + 6 · 0 · 0 −− 6 · 2 · 0 − 4 · 0 · 3 − 1 · 5 · 0 = 1 · 2 · 3 = 6.General definitionThe general definition of the determinant is quitecomplicated as there is no simple explicit formula.There are several approaches to defining determinants.Approach 1 (original): an explicit (but verycomplicated) formula.Approach 2 (axiomatic): we formulateproperties that the determinant should have.Approach 3 (inductive): the determinant of ann×n matrix is defined in terms of determinants ofcertain (n − 1)×(n − 1) matrices.Mn(R): the set of n×n matrices with real entries.Theorem There exists a unique functiondet : Mn(R) → R (called the determinant) with thefollowing 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 scalarto another row, the determinant remains the same;• if we interchange two rows of a matrix, thedeterminant changes its sign;• det I = 1.Corollary det A = 0 if and only if the matrix A issingular.Row echelon form of a square matrix A:∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗det A 6= 0 det A = 0Example. A =3 −2 01 0 1−2 3 0, det A =?In the previous lecture we have transformed thematrix A into the identity matrix using elementaryrow operations.• interchange the 1st row with the 2nd row,• add −3 times the 1st row to the 2nd row,• add 2 times t he 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 t he 3rd row to the 2nd row,• add −1 times the 3rd row to the 1st row.Example. A =3 −2 01 0 1−2 3 0, det A =?In the previous lecture we have transformed thematrix A into the identity matrix using elementaryrow operations.These included two row multiplications, by −0.5and by −0.4, and one row exchange.It follows thatdet 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 identical rows thendet A = 0.a1a2a3b1b2b3a1a2a3= 0• If a matrix A has two rows proportional thendet A = 0.a1a2a3b1b2b3ra1ra2ra3= ra1a2a3b1b2b3a1a2a3= 0Distributive law for rows• Suppose that matrices X , Y , Z are identicalexcept for the ith row and the ith row of Z is thesum of the ith rows of X and Y .Thendet Z = det X + det Y .a1+a′1a2+a′2a3+a′3b1b2b3c1c2c3=a1a2a3b1b2b3c1c2c3+a′1a′2a′3b1b2b3c1c2c3• Adding a scalar multiple of one row to anotherrow does not change the determinant of a matrix.a1+ rb1a2+ rb2a3+ rb3b1b2b3c1c2c3==a1a2a3b1b2b3c1c2c3+rb1rb2rb3b1b2b3c1c2c3=a1a2a3b1b2b3c1c2c3Definition. A square matrix A = (aij) is calledupper triangular if all entries below the maindiagonal are zeros: aij= 0 whenever i > j.• The determinant of an upper triangular matrix isequal to the product of its diagonal entries.a11a12a130 a22a230 0 a33= a11a22a33• If A = diag(d1, d2, . . . , dn) thendet A = d1d2. . . dn. In particular, det I = 1.Determinant of the transpose• If A is a square matrix then det AT= det A.a1b1c1a2b2c2a3b3c3=a1a2a3b1b2b3c1c2c3Columns vs. rows• If one column of a matrix is multiplied by ascalar, the determinant is multiplied by the samescalar.• Interchanging two columns of a matrix changesthe sign of its determinant.• If a matrix A has two columns proportional thendet A = 0.• Adding a scalar multiple of one column toanother does not change the determinant of amatrix.SubmatricesDefinition.
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