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Cofactor Expansions the Determinant Is a Scalar Characteristic of a Square

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Cofactor Expansions the Determinant Is a Scalar Characteristic of a Square Linear Algebra Grinshpan Cofactor expansions The determinant is a scalar characteristic of a square matrix. The determinant of A is denoted by det A or by jAj: For 1 × 1; 2 × 2; and 3 × 3 matrices the formulas are: [ ] det a = a; [ ] a b det = ad − bc; c d 2 3 a b c det 4d e f5 = aei + bfg + cdh − ceg − bdi − afh: g h i What is the pattern? One way to describe it is as follows. i+j Let A be n × n: To each entry aij of A; associate the sign (−1) and the submatrix Aij obtained from A by deleting the i-th row and the j-th column. i+j The number (−1) jAijj is the cofactor of A corresponding to the position (i; j): For every row index i; i+1 i+2 i+n jAj = ai1(−1) jAi1j + ai2(−1) jAi2j + ::: + ain(−1) jAinj: This is the Laplace or cofactor expansion of jAj along the i-th row. For every column index j; 1+j 2+j n+j jAj = a1j(−1) jA1jj + a2j(−1) jA2jj + ::: + anj(−1) jAnjj: This is the cofactor expansion of jAj along the j-th column. The above expansions give jAj in terms of determinants of one order less, so any of them may be used for calculating jAj: The reduction may take several steps. 1 2 3 2 3 1 3 1 2 4 5 6 = −4 + 5 − 6 = −4(18 − 24) + 5(9 − 21) − 6(8 − 14) = 0 8 9 7 9 7 8 7 8 9 1 2 3 5 6 2 3 2 3 4 5 6 = +1 − 4 + 7 = (45 − 48) − 4(18 − 24) + 7(12 − 15) = 0 8 9 8 9 5 6 7 8 9 For every 2 × 2 matrix, the determinant is plus or minus the area of the parallelogram bounded by its columns (rows). For every 3 × 3 matrix, the determinant is plus or minus the volume of the parallelepiped bounded by its columns (rows). For instance, the columns of the preceding determinant are coplanar, so enclose zero volume. Properties of the determinant The determinant may be defined by the following four properties. (1) Normalization: the determinant of an identity matrix is 1 [ ] 1 0 det = 1 0 1 (2) Antisymmetry: a column interchange changes the sign of the determinant 2 3 2 3 a ∗ d d ∗ a det 4b ∗ e5 = − det 4e ∗ b5 c ∗ f f ∗ c (3) Column additivity [ ] [ ] [ ] ∗ a + b ∗ a ∗ b det = det + det ∗ c + d ∗ c ∗ d (4) Column homogeneity 2 3 2 3 ∗ 3a ∗ ∗ a ∗ det 4∗ 3b ∗5 = 3 det 4∗ b ∗5 ∗ 3c ∗ ∗ c ∗ Immediate consequences: A matrix with two identical columns has zero determinant. Adding a multiple of one column to another does not change the determinant. A matrix with a zero column has zero determinant. The determinant of a permutation matrix is 1: The above set of rules allows us to calculate any determinant. For instance, using additivity and homogeneity in each column, we have 2 5 2 5 0 5 = + 3 7 0 7 3 7 2 5 2 0 0 5 0 0 = + + + 0 0 0 7 3 0 3 7 1 1 1 0 0 1 0 0 = 2 · 5 + 2 · 7 + 3 · 5 + 3 · 7 0 0 0 1 1 0 1 1 = 2 · 5 · 0 + 2 · 7 · (+1) + 3 · 5 · (−1) + 3 · 7 · 0 = −1: Similarly, 1 2 3 1 0 0 1 0 0 0 2 0 0 0 3 0 0 3 0 2 0 4 5 6 = 0 5 0 + 0 0 6 + 4 0 0 + 4 0 0 + 0 5 0 + 0 0 6 7 8 9 0 0 9 0 8 0 0 0 9 0 8 0 7 0 0 7 0 0 =1 · 5 · 9 · (+1) + 1 · 6 · 8 · (−1) + 2 · 4 · 9 · (−1) + 3 · 4 · 8 · (+1) + 3 · 5 · 7 · (−1) + 2 · 6 · 7 · (+1) = 0: We see that the determinant is a sum of signed products. In each product the entries are taken one per column and one per row. The sign is given by the determinant of the corresponding permutation matrix. For an n × n matrix, there are n! summands. For any permutation matrix P; we have det P = det P >: Therefore, transposition does not change the determinant, det A = det A>; and all of the above properties equally well apply to rows. A row interchange changes the sign of the determinant [ ] [ ] 0 1 1 0 det = − det 1 0 0 1 The determinant is row additive [ ] [ ] [ ] 1 2 0 1 1 1 det = det + det a b a b a b The determinant is row homogeneous [ ] [ ] 1 3 1 3 det = 2 det 2 4 1 2 A matrix with two identical rows has zero determinant. Adding a multiple of one row to another does not change the determinant. A matrix with a zero row has zero determinant. Here are some more practical observations. The determinant of a diagonal matrix is the product of its diagonal entries. The same is true for triangular matrices. 2 3 2 3 2 3 a 0 0 a ∗ ∗ a 0 0 det 40 b 05 = det 40 b ∗5 = det 4∗ b 05 = abc 0 0 c 0 0 c ∗ ∗ c The determinant and row operations We now know what happens to the determinant under elementary row operations. Row interchange: the determinant changes sign 2 3 2 3 1 2 3 4 5 6 det 44 5 65 = − det 41 2 35 ∗ ∗ ∗ ∗ ∗ ∗ Scale a row: scale the determinant 2 3 2 3 ∗ ∗ ∗ ∗ ∗ ∗ det 43a 3b 3c5 = 3 det 4a b c5 ∗ ∗ ∗ ∗ ∗ ∗ Add a multiple of one row to another: no change 2 3 2 3 a b c a b c det 4d e f5 = det 4d + 2a e + 2b f + 2c5 ∗ ∗ ∗ ∗ ∗ ∗ Based on these properties are several useful conclusions. If A is reduced to an echelon form U using k row interchanges and no scalings, then det A = (−1)k det U = (−1)k × (product of diagonal entries of U): So a matrix is invertible if and only if its determinant is nonzero. A matrix with linearly dependent rows and columns has zero determinant. 2 3 1 2 3 det 44 5 65 = 0 7 8 9 The determinant is multiplicative: det(AB) = (det A)(det B): If A is invertible, then det(A−1) = 1= det A: Exercises Calculate the determinants 3 1 1 1 1 1 1 1 2 0 0 3 0 1 1 x 1 1 1 2 4 1 3 1 1 1 −1 1 −1 0 2 3 0 1 0 1 ; ; 1 x 1 ; ; 1 3 9 ; : 1 1 3 1 1 1 −1 −1 0 3 2 0 1 1 0 1 1 x 1 5 25 1 1 1 3 1 −1 −1 1 3 0 0 2 .
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