ENGG2013 Unit 9 3x3 Determinant Feb, 2011.

Last time 22 determinant Compute the area of a parallelogram by determinant A formula for 2x2 matrix inverse kshum ENGG2013

Today 33 determinant and its properties Using determinant, we can test whether three vectors lie on the same plane solve 33 linear system test whether the inverse of a 33 matrix exists kshum ENGG2013

Vector Notation We will use two different notations for a point in the 3D space (x,y,z) z z y y x x kshum ENGG2013

22 determinant Notation for 22 determinant : - bc – ad + How to calculate: ad – bc kshum ENGG2013

33 determinant Notation for 33 determinant : Definition: kshum ENGG2013

Rule of Sarrus – – + + – + Pierre Frédéric Sarrus (1798 – 1861) \begin{array}{|ccc|cc} a_1 & b_1 & c_1 & {\color{blue} a_1} & {\color{blue} b_1}\\ a_2 & b_2 & c_2 & {\color{blue} a_2} &{\color{blue} b_2}\\ a_3 & b_3 & c_3 & {\color{blue} a_3} &{\color{blue} b_3}\\ \end{array} Pierre Frédéric Sarrus (1798 – 1861) kshum ENGG2013

Volume of parallelepiped Geometric meaning The magnitude of 33 determinant is the volume of a parallelepiped z y x kshum ENGG2013

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

Det of Diagonal matrix Volume of a rectangular box c b a kshum ENGG2013

Transpose has the same determinant – – + + + – Compare with kshum ENGG2013

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

Question Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on the same plane? kshum ENGG2013

Cramer’s rule If the determinant of a 33 matrix A is non-zero, we can solve the linear system A x = b by Cramer’s rule. The solution to is or equivalently A x b \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} x_1\\ x_2\\ x_3 = b_1\\ b_2\\ b_3 Gabriel Cramer (1704-1752) kshum ENGG2013

PROPERTIES OF DETERMINANT kshum ENGG2013

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

Properties of determinant Taking transpose does not change the value of determinant We have already verified this property in p.11 kshum ENGG2013

Meta-property Because After taking the transpose of a matrix, columns become rows, and rows become column. 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

Properties of determinant If any row or column is zero, then the determinant is 0. For example kshum ENGG2013

Properties of determinant 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

Properties of Determinant 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

Multiply by a constant If we multiply a row (or a column) by a constant k, the value of determinant increases by a factor of k. For example, if we multiply the third row by a constant k, The 2nd kind of elementary row operation kshum ENGG2013

An additive property 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

Properties of Determinant If we add a constant multiple of a row (column) to the other row (column), the determinant does not change. For example, if we replace the 3rd column by the sum of the 3rd column and k times the 2nd column, The 3rd kind of elementary row operation kshum ENGG2013

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

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. Etc. \begin{vmatrix} a_{1} & b_1 & c_{1}+kb_1 \\ a_2 & b_2 & c_2 +kb_2\\ a_{3} & b_3 & c_{3}+kb_3 \end{vmatrix} = a_{1} & b_1 & c_{1} \\ a_2 & b_2 & c_2 \\ a_{3} & b_3 & c_{3} Cramer’s rule in wikipedia kshum ENGG2013

Because x1, x2, x3 satisfy the system of linear equations, we have By substitution Property 6 \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} + x_2 a_{12}& a_{12} & a_{13}\\ a_{22} & a_{22} & a_{23}\\ a_{32}& a_{32} & a_{33} \end{vmatrix}+ x_3 a_{13} & a_{12} & a_{13}\\ a_{23} & a_{22} & a_{23}\\ a_{33} & a_{32} & a_{33} Property 5 =0 By Property 3 kshum ENGG2013

MINOR AND COFACTOR kshum ENGG2013

Another way to compute det Group the six terms as 33 determinant can be computed in terms of 22 determinant kshum ENGG2013

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

Expansion on the first row \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}=a_{1}\begin{vmatrix} b_2 & c_2\\ b_3 & c_3 \end{vmatrix} - b_1\begin{vmatrix} a_2 & c_2\\ a_3 & c_3 \end{vmatrix}+c_1 a_2 & b_2\\ a_3 & b_3 \end{vmatrix} \\ Minor of a1 Minor of b1 Minor of c1 kshum ENGG2013

Expansion on the second row \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}= -a_2\begin{vmatrix} b_1 & c_1\\ b_3 & c_3 \end{vmatrix} + b_2\begin{vmatrix} a_1 & c_1\\ a_3 & c_3 \end{vmatrix} - c_2 a_1 & b_1\\ a_3 & b_3 \end{vmatrix} \\ Minor of a2 Minor of b2 Minor of c2 kshum ENGG2013

Expansion on the third row \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}= a_3\begin{vmatrix} b_1 & c_1\\ b_2 & c_2 \end{vmatrix} - b_3\begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix} + c_3 a_1 & b_1\\ a_2 & b_2 \end{vmatrix} \\ Minor of a3 Minor of b3 Minor of c3 kshum ENGG2013

The sign pattern Expansion on the first row Expansion on the second row Expansion on the last row kshum ENGG2013

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

Cofactor The minor together with the appropriate  sign is called cofactor. For The cofactor of Cij is defined as Expansion on the i-th row (i=1,2,3): Expansion on the j-th column (j=1,2,3): The sign The minor of aij kshum ENGG2013

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

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