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ENGG2013 Unit 9 3x3 Determinant Feb, 2011. Last time • 22 determinant • Compute the area of a parallelogram by determinant • A formula for 2x2 matrix inverse kshum ENGG2013 2 Today • 33 determinant and its properties • Using determinant, we can – test whether three vectors lie on the same plane – solve 33 linear system – test whether the inverse of a 33 matrix exists kshum ENGG2013 3 Vector Notation • We will use two different notations for a point in the 3D space (x,y,z) z z y y x kshum x ENGG2013 4 22 determinant Notation for 22 determinant : + – How to calculate: ad – bc - bc kshum ENGG2013 ad 5 33 determinant Notation for 33 determinant : Definition: kshum ENGG2013 6 Rule of Sarrus + + + – – – Pierre Frédéric Sarrus (1798 – 1861) kshum ENGG2013 7 Volume of parallelepiped • Geometric meaning – The magnitude of 33 determinant is the volume of a parallelepiped z y x kshum ENGG2013 8 Co-planar zero determinant • Determinant = 0 Volume = 0 the three vectors lie on the same plane z y A collection of vectors are said to be co-planar if they lie on the same plane. x kshum ENGG2013 9 Det of Diagonal matrix • Volume of a rectangular box c b a kshum ENGG2013 10 Transpose has the same determinant + + + – – – Compare with kshum ENGG2013 11 Volume of parallelepiped • In computing the volume of a parallelepiped, it does not matter whether we write the vector horizontally or vertically in the determinant z Volume of parallelepiped with vertices (0,0,0), (1,2,0), (2,0,1), (–1, 1, 3) equals to the absolute value of y or x kshum ENGG2013 12 Question • Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on the same plane? kshum ENGG2013 13 Cramer’s rule • If the determinant of a 33 matrix A is non-zero, we can solve the linear system A x = b by Cramer’s rule. • The solution to or equivalently A is x b Gabriel Cramer (1704-1752) kshum ENGG2013 14 PROPERTIES OF DETERMINANT kshum ENGG2013 15 How to show that Cramer’s rule does give the correct answer? • The Cramer’s rule is a theorem, which requires a proof, or verification. • We need some properties of determinant. kshum ENGG2013 16 Properties of determinant 1. Taking transpose does not change the value of determinant We have already verified this property in p.11 kshum ENGG2013 17 Meta-property • Because 1. After taking the transpose of a matrix, columns become rows, and rows become column. 2. Taking the transpose of a matrix does not change the value of its determinant. • Therefore, any row property of determinant is automatically a column property, and vice versa. kshum ENGG2013 18 Properties of determinant 2. If any row or column is zero, then the determinant is 0. For example kshum ENGG2013 19 Properties of determinant 3. If any two columns (or rows) are the identical, then the determinant is zero. For example, if the second column and the third column are the same, then kshum ENGG2013 20 Properties of Determinant 4. If we exchange of the two columns (or two rows), the determinant is multiplied by –1. For example, if we exchange the column 2 and column 3, we have The first kind of elementary row operation kshum ENGG2013 21 Multiply by a constant 5. If we multiply a row (or a column) by a constant k, the value of determinant increases by a factor of k. The 2nd kind of elementary row operation For example, if we multiply the third row by a constant k, kshum ENGG2013 22 An additive property 6. If a row (or column) of a determinant is the sum of two rows (or columns), the determinant can be split as the sum of two determinants For example, if the first column is the sum of two column vectors, then we have kshum ENGG2013 23 Properties of Determinant 7. If we add a constant multiple of a row (column) to the other row (column), the determinant does not change. The 3rd kind of elementary row operation For example, if we replace the 3rd column by the sum of the 3rd column and k times the 2nd column, kshum ENGG2013 24 Summary on the effect of the elementary row (or column) operations on determinant • Exchange two rows (or columns) change the sign of determinant • Multiply a row (or a column) by a constant k multiply the determinant by k • Add a constant multiple of a row (column) to another row (or column) no change kshum ENGG2013 25 Proof of the Cramer’s rule • The solution to is Verification for x1: Substitute the value of b1, b2 and b3 in the first column of A. Verification for x2: Substitute the value of b1, b2 and b3 in the second column of A. Cramer’s rule in wikipedia Etc. kshum ENGG2013 26 Because x1, x2, x3 satisfy the system of linear equations, we have By substitution Property 6 Property 5 =0 kshum ENGG2013 =0 By Property 3 27 MINOR AND COFACTOR kshum ENGG2013 28 Another way to compute det Group the six terms as 33 determinant can be computed in terms of 22 determinant kshum ENGG2013 29 Minor and cofactor • A minor of a matrix is the determinant of some smaller square matrix, obtained by removing one or more of its rows and columns. • Notation: Given a matrix A, the minor obtained by removing the i-th row and j-th column is denoted by Aij. It is also called the minor of the (i,j)-entry aij in A. kshum ENGG2013 30 Expansion on the first row Minor of a1 kshum ENGG2013 Minor of b1 Minor of c1 31 Expansion on the second row Minor of a2 kshum ENGG2013 Minor of b2 Minor of c2 32 Expansion on the third row Minor of a3 kshum ENGG2013 Minor of b3 Minor of c3 33 The sign pattern Expansion on the first row Expansion on the second row Expansion on the last row kshum ENGG2013 34 Column expansion We have similar recursive formula for determinant by column expansion For example, Computation on the third column is easy, because there are lots of zeros. kshum ENGG2013 35 Cofactor • The minor together with the appropriate sign is called cofactor. • For The sign The cofactor of Cij is defined as Expansion on the i-th row (i=1,2,3): The minor of aij Expansion on the j-th column (j=1,2,3): kshum ENGG2013 36 A formula for matrix inverse Suppose that det A is nonzero. (Beware of the subscripts) Usually called the adjoint of A Three steps in computing above formula 1. for i,j = 1,2,3, replace each aij by cofactor Cij 2. Take the transpose of the resulting matrix. 3. divide by the determinant of A. kshum ENGG2013 37 A Quotation Algebra is but written geometry; geometry is but drawn algebra. --- Sophie Germain (1776-1831) L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée kshum ENGG2013 38