MATH 311-504Topics in Applied MathematicsLecture 11:Properties of determinants.Determinant is a scalar assigned to each square matrix.Notation. The determinant of a matrixA = (aij)1≤i,j≤nis denoted det A ora11a12. . . a1na21a22. . . a2n............an1an2. . . ann.Principal property: det A = 0 if and only if thematrix A is not invertible.Definition in low dimensionsDefinition. det (a) = a,a bc d= ad − bc,a11a12a13a21a22a23a31a32a33= a11a22a33+ a12a23a31+ a13a21a32−−a13a22a31− a12a21a33− a11a23a32.+ :*∗ ∗∗ *∗∗ ∗ *,∗ *∗∗ ∗ ** ∗ ∗,∗ ∗ ** ∗ ∗∗ *∗.− :∗ ∗ *∗ * ∗*∗ ∗,∗ *∗*∗ ∗∗ ∗ *,*∗ ∗∗ ∗ *∗ * ∗.Examples: 3×3 matrices3 −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 are no simple explicit formula.There are several approaches to defining determinants.Approach 1 (original): an ex plicit (but verycomplicated) formula.Appr oach 2 (axiomatic): we formulateproperties that the determinant should have.Appr oach 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 multipl ied 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 onl y if the matrix A isnot invertible.Example. A =3 −2 01 0 1−2 3 0, det A =?In the previous lecture we have transformed thematrix A into the identity matri x using elementaryrow 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 −1/2,• add −3 times the 2nd row to the 3rd row,• multiply the 3rd row by −2/5,• add −3/2 times the 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 matri x using elementaryrow operations.These included two row multiplications, by −1/2and by −2/5, and o ne row exchange.It follows thatdet I = −−12−25det A = −15det A.Hence det A = −5 det I = −5.Other pro perties of determinants• If a matrix A has two identi cal rows thendet A = 0.a1a2a3b1b2b3a1a2a3= 0• If a matrix A has two rows proportional thendet A = 0.a1a2a3b1b2b3ra1ra2ra3= ra1a2a3b1b2b3a1a2a3= 0Distributive law for rows• Suppose that matrices A, B, C are identicalexcept for the i th row and the i th row of C is thesum of the i th rows of A and B.Then det A = det B + det C .a1+a′1a2+a′2a3+a′3b1b2b3c1c2c3=a1a2a3b1b2b3c1c2c3+a′1a′2a′3b1b2b3c1c2c3• Adding a scalar multiple o f one row to anotherrow does not change the determinant of a matrix.a1+ rb1a2+ rb2a3+ rb3b1b2b3c1c2c3==a1a2a3b1b2b3c1c2c3+rb1rb2rb3b1b2b3c1c2c3=a1a2a3b1b2b3c1c2c3Definition. A square matrix A = (aij) is calledupper triangular if all entries below the maindiagonal are zer os: aij= 0 whenever i > j.• The determinant o f an upper triangular matrix i sequal 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.Definition. Given a matrix A, the transpose of A,denoted ATor At, is the matrix obtained byinterchanging rows and columns in the matrix A.That is, if A = (aij) then AT= (bij), where bij= aji.Example.a1a2a3b1b2b3T=a1b1a2b2a3b3.• If A is a square matrix then det AT= det A.a1b1c1a2b2c2a3b3c3=a1a2a3b1b2b3c1c2c3Columns v s. 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 o f one column toanother does not change the determinant of amatrix.SubmatricesDefinition. Given a matrix A, a k×k submatrix ofA is a matrix obtained by specifying k columns andk rows of A and deleting the other columns androws.1 2 3 410 20 30 403 5 7 9→∗ 2 ∗ 4∗ ∗ ∗ ∗∗ 5 ∗ 9→2 45 9If A is an n×n matrix then Aijdenote the(n − 1)×(n − 1) submatrix obtained by deleting theith row and the jth column.Example. A =3 −2 01 0 1−2 3 0.A11=0 13 0, A12=1 1−2 0, A13=1 0−2 3,A21=−2 03 0, A22=3 0−2 0, A23=3 −2−2 3,A31=−2 00 1, A32=3 01 1, A33=3 −21 0.Row and column expansionsTheorem Let A be an n×n matrix. Then for any1 ≤ k, m ≤ n we have thatdet A =nXj=1(−1)k+jakjdet Akj,(expansion by kth row)det A =nXi=1(−1)i+maimdet Aim.(expansion by mth column)Signs for row/column expansions+ − + − · · ·− + − + · · ·+ − + − · · ·− + − + · · ·...............Example. A =3 −2 01 0 1−2 3 0.Expansion by the 1st row:3 −2 01 0 1−2 3 0= 30 13 0− (−2)1 1−2 0=