Determinants in Linear Algebra - Properties, Formulas, and Applications
Explore the world of determinants in linear algebra through concepts like properties, formulas, Cramer's rule, and more. Dive into the significance of determinants in matrix operations, invertibility, and applications in high school mathematics.
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Determinant Hung-yi Lee
Reference MIT OCW Linear Algebra: Lecture 18: Properties of determinants http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra- spring-2010/video-lectures/lecture-18-properties-of-determinants/ Lecture 19: Determinant formulas and cofactors http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra- spring-2010/video-lectures/lecture-19-determinant-formulas-and- cofactors/ Lecture 20: Cramer's rule, inverse matrix, and volume http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra- spring-2010/video-lectures/lecture-20-cramers-rule-inverse-matrix- and-volume/ Textbook: Chapter 3
Determinant The determinant of a square matrix is a scalar that provides information about the matrix. E.g. Invertibility of the matrix. Learning Target The formula of Determinants The properties of Determinants Cramer s Rule
Formula for Determinants
Determinants in High School 2 X 2 3 x 3 ?1 ?4 ?7 ?2 ?5 ?8 ?3 ?6 ?9 ? =? ? ? ? = ? ??? ? = ??? ? = ?? ?? ?1?5?9+?2?6?7+?3?4?8 ?3?5?7 ?2?4?9 ?1?6?8
Cofactor Expansion Suppose A is an n x n matrix. Aijis defined as the submatrix of A obtained by removing the i-th row and the j-th column. A i-th row J-th column
Cofactor Expansion Pick row 1 ???? = ?11?11+ ?12?12+ + ?1??1? Or pick row i ???? = ??1??1+ ??2??2+ + ?????? cij: (i,j)-cofactor Or pick column j ???? = ?1??1?+ ?2??2?+ + ?????? ???= 1?+??????? Cofactor expansion again
2 x 2 matrix ???= 1?+??????? Define det([a]) = a ? =? ? ? ??? ? = ?? ?? ? Pick the first row ??? ? = ??11+ ??12 ?11= 11+1??? ? = ? ?12= 11+2??? ? = ?
3 x 3 matrix ???= 1?+??????? 1 4 7 2 5 8 3 6 9 ? = Pick row 2 ???? = ?21?21+ ?22?22+ ?23?23 4 6 5 12+3????23 12+1????21 12+2????22 1 4 7 2 5 8 3 6 9 1 4 7 2 5 8 3 6 9 1 4 7 2 5 8 3 6 9 ?21= ?22= ?23=
Example Given tridiagonal n x n matrix A 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 ? = Find detA when n = 999
????4 = ?11?11+ ?12?12+ ?13?13+ ?14?14 1 1 ?2=1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 ?11= 12??? ?3= = ??? ?3 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 ?4= ?12= 13??? = ??? ?2 = ?11?11+ ?12?12+ ?13?13 1 1 0 = ??? ?2
Example ??? ?4 = ??? ?3 ??? ?2 ??? ?? = ??? ?? 1 ??? ?? 2 ??? ?1 = 1 ??? ?3 = 1 ??? ?2 = 0 ??? ?5 = 0 ??? ?6 = 1 ??? ?4 = 1 ??? ?7 = 1 ??? ?8 = 0
Properties of Determinants Volume in high dimensions (?)
Determinants in High School ?1 ?4 ?7 ?2 ?5 ?8 ?3 ?6 ?9 2 X 2 3 x 3 ? = ? =? ? ? ? ?7,?8,?9 |??? ? | ! (c,d) V (a,b) ?1,?2,?3 ?4,?5,?6
Three Basic Properties Basic Property 1: ??? ? = 1 Basic Property 2: Exchange rows only reverses the sign of det (do not change absolute value) Basic Property 3: Determinant is linear for each row Area in 2d and Volume in 3d have the above properties Can we say determinant is the Volume also in high dimension?
Three Basic Properties Basic Property 1: ??? ? = 1 1 1 1 0 0 0 1 0 0 0 1 ?2=1 0 1 ?3= 0 ??? ?2 = 1 ??? ?3 = 1
Three Basic Properties Basic Property 2: Exchanging rows only reverses the sign of det 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 ??? = 1 1 0 0 1 ??? = 1 ??? = 1 0 1 1 0 ??? = 1 ??? = 1
Three Basic Properties Basic Property 2: Exchanging rows only reverses the sign of det If a matrix A has 2 equal rows ??? ? = 0 exchange two rows ? ? ??? ? = ? = ??? ? = ? Exchanging the two equal rows yields the same matrix
Three Basic Properties Basic Property 3: Determinant is linear for each row 3-a ?? ? ?? ? ? ? ? ? ??? = ???? (c,d) (c,d) V V V (2a,2b) (a,b)
Three Basic Properties Basic Property 3: Determinant is linear for each row 3-a ?? ? ?? ? ? ? ? ? ??? = ???? Q: find ??? 2? If A is n x n A:??? 2? = 2???? ?
Three Basic Properties Basic Property 3: Determinant is linear for each row 3-a ?? ? ?? ? ? ? ? ? ??? = ???? A row of zeros Set ? = 0! ??? ? = 0 A row of zeros volume is zero
Three Basic Properties Basic Property 3: Determinant is linear for each row ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? 3-b ??? = ??? + ??? (c,d) (c,d) (a+a ,b+b ) (a+a ,b+b ) (a,b) (a,b)
Three Basic Properties Basic Property 3: Determinant is linear for each row Subtract k x row i from row j (elementary row operation) Determinant doesn t change ? ? ??? ? ?? ? ?? ? ? ? ? ? ? 3-b = ??? + ??? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? 3-a = ??? ???? = ???
Determinants for Upper Triangular Matrix ?1 0 Killing everything above ? = Does not change the det ?? ?1 0 0 ?? ??? ? = ??? Property 1 1 0 = 1 0 1 3-a = ?1?2 ????? ??? ? = ?1?2 ??(Products of diagonal)
Determinants v.s. Invertible A is invertible det(A) 0 A R Elementary row operation ??? ? ??? ? = ?1?2 ??? ? If A is invertible, R is identity Exchange: Change sign ??? ? 0 ??? ? = 1 Scaling: Multiply k R has zero row If A is not invertible, nothing Add row: ??? ? = 0 ??? ? = 0
We collect one more properties for invertible! Invertible Let A be an n x n matrix. A is invertible if and only if The columns of A span Rn For every b in Rn, the system Ax=b is consistent The rank of A is n The columns of A are linear independent The only solution to Ax=0 is the zero vector The nullity of A is zero The reduced row echelon form of A is In A is a product of elementary matrices There exists an n x n matrix B such that BA = In There exists an n x n matrix C such that AC = In det(A) 0 onto One- on-one
Example A is invertible det(A) 0 1 1 0 1 2 ? 7 For what scalar c is the matrix not invertible? ? = 1 2 det(A) = 0 +2 1 1 ???? = 1 0 7 + 1 ? 2 2 0 2 1 1 7 1 ? 1 = 0 2? 2 7 ? = 3? 9 3? 9 = 0 ? = 3 not invertible
More Properties of Determinants ??? ? + ? ??? ? + ??? ? ??? ?? = ??? ? ??? ? Q: find ??? ? 1 ? 1? = ? ??? ? 1??? ? = ??? ? = 1 ??? ? 1= 1 ??? ? Q: find ??? ?2 ??? ?2 = ??? ?2 = ??? ? ??? ? ??? ??= ??? ? Zero row zero column Same row same column P212 - 215
Formula for A-1 ?11 ??1 ?1? ??? 1 ? 1= ??? ? : scalar ?: cofactors of A (? has the same size as A, so does ??) ?? is adjugate of A (adj A, ) ??? ??? ? = ?11 ?21 ? ? ? ? ?12 ?22 ? ? ? ? ? = ? =? ? ? ? 1 ? = 1 ? ? ? ??? ? = ? ?? ?? ??= = ?? ??
1 Formula for A-1 ? 1= ??? ??? ? ? ? ? ? ? ? ? ,? 1=? ? = ??? ? = ??? + ??? + ?? ??? ??? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? = + ? ? ? ? ? ? ? ? ? ? ? ? + +
1 Formula for A-1 ? 1= ??? ??? Proof: ???= ??? ? ?? ?11 ??1 ?1? ??? ?11 ?1? ??1 ??? ??? ? 0 = 0 ??? ? transpose Diagonal: By definition of determinants Not Diagonal: (Exercise 82, P221)
1 Cramer s Rule ? 1= ??? ??? ?1=??? ?1 ??? ? ?1=with column 1 replaced by b ?? = ? ?2=??? ?2 ??? ? ? = ? 1? n-1 Columns of A ? 1 ??? ???? = ??=??? ?? ??? ? ??=with column j replaced by b
Formula from Three Properties 2 1 1 0 0 1 0 1 1 0 = 1 = 1 = det? 0 ? det? ? ? +det0 ? ? 3-b ? ? ? = det? 0 0+ det? 3-a 0 ? +det0 ? 0+ det0 3-a ? ? 3-b ? 0 3-a ? 0 3-a = ?? = bc = 0 = 0 = ?? ??
Finally, we get 3 x 3 x 3 matrices Most of them have zero determinants ?11 ?21 ?31 ?12 ?22 ?32 ?13 ?23 ?33 det ?11 ?21 ?31 0 0 0 ?12 ?22 ?32 0 0 0 ?13 ?23 ?33 ?22 ?32 ?23 ?33 ?21 ?31 ?23 ?33 ?21 ?31 ?22 ?32 = det +det +det ?11 0 ?31 0 0 0 ?11 ?21 ?31 0 0 0 0 ?11 0 ?31 0 0 0 ?22 ?32 +??? = ??? ?23 ?33 +??? ?33 ?32 ?33 ?32 ?11 ?21 ?31 0 0 0 0 0 0 ?11 ?21 0 0 0 0 0 0 ?11 ?21 0 0 0 0 0 0 = ??? +??? +??? ?32 ?33
?11 ?21 ?31 ?12 ?22 ?32 ?13 ?23 ?33 3! matrices have non-zero rows det 0 ?12 0 0 0 0 ?11 0 0 ?11?22?33 0 0 0 ?11 0 0 ?11?23?32 0 0 0 ?21 0 ?12?21?33 + ?22 0 = ?23 0 + ?33 ?33 ?32 0 0 ?12 0 0 0 0 0 0 ?13 0 0 0 0 0 ?13 0 0 ?23 0 + ?21 0 ?13?21?32 + ?22 0 + ?31 ?32 ?31 ?13?22?31 ?12?23?31 Pick an element at each row, but they can not be in the same column.
Formula from Three Properties Given an n x n matrix A ??? ? = ?!????? ?1??2??3? ??? Format of each term: Find an element in each row permutation of 1,2, , n
Example 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 ??? 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 = ??? +??? +1 -1
Formulas for Determinants ???? = ?!????? ?1??2??3? ??? Format of each term: ???? = ?11?11+ ?12?12+ + ?1??1? All terms including ?11 All terms including ?12 All terms including ?1?
