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# Class Note for MATH 410 with Professor Dostert at UA

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Date Created: 02/06/15

THE UNIVERSITY a OF ARIZONA Math 410 Matrix Analysis Section 18 General Linear Systems Section 19 Determinants Paul Dostert January 14 2009 113 A19 Row Echelon Form An m gtlt n matrix A is in row echelon form ref if a Any rows with all zeros are at the bottom b The leading entry in each nonzero row is in a column to the left of the rows below it Ex The following are in row echelon form 1 2 3 4 4 5 17 1 1 0 2 0 8 11 6 0 9 3 7 0 3 3 2 7 0 0 1 2 0 0 4 0 0 9 11 0 0 0 6 0 0 0 0 Prop Any matrix can be reduced to ref by a sequence of elementary matrix operations of type 1 and 2 This is the same as saying if A is an m gtlt n matrix then we can write PA LU where U is in ref with PL m gtlt m matrices A Rank The nonzero entries that begin each row of the ref are called pivots For example if we have a 4 gtlt 5 matrix with 2 pivots then 2 of the rows in the ref are exactly zero The rank of a matrix A is the number of pivots Note that this would also correspond to the number of equations left after reducing the system to ref Prop A square matrix of size n gtlt n is nonsingular iff its rank is equal to n In a system reduced by GE where Um c in ref the variables corresponding to columns containing a pivot are called basic variables sometimes leading variables while the variables corresponding to the columns without a pivot are called free variables Ex Find the basic amp free variables for each of the following 1 2 3 4 4 5 17 1 1 0 2 0 8 11 6 0 9 3 0 3 3 2 0 0 1 2 0 0 4 0 0 9 11 0 0 0 6 0 0 0 0 A Solving the System Ax b Thm A system Ax b of m linear equations in n unknowns has either a Exactly one solution b infinitely many solutions or c no solution inconsistent Ex Determine the number of solutions to each of the systems xy a xy xy b xy xy 1 2x2y Ol Ol l C l l D What do each of the above systems represent graphically A11 Homogeneous Systems and Examples A system of linear equations is homogeneous if each constant term is zero if the RHS is zero Thm A homogeneous linear system Ax 0 of m equations in n unknowns has a non trivial solution x 7 0 iff rankA 7quot lt n If m lt n the system always has a non trivial solution If m n the system has a non trivial solution iff A is singular Ex Find the intersection of the planes c y z 0 and c 2y z 0 Ex Find all solutions to 0 1 2 x1 0 1 0 1 x2 0 1 2 5 x3 0 At Determinant of a 2 x 2 Matrix Recall we have previously defined the determinant of a 2 gtlt 2 matrix A all Q12 as det A a11a22 a12a21 How was this used in the G21 G22 formula for the inverse of a 2 gtlt 2 matrix The determinant of any matrix A is usually denoted by either A or det A We may also write 011 012 A l a a l 21 22 We essentially replace the bracket notation around the matrix with absolute value type notation indicated we want the determinant of the matrix We define the determinant of a 1 gtlt 1 matrix A as detA a a where a does NOT indicate absolute value At Determinant of a 3 x 3 Matrix The determinant of a 3 gtlt 3 matrix is a linear combination of determinants of 2 gtlt 2 matrices The process is like the cross product G11 G12 G13 Let A a21 a22 a23 Then the determinant of A is the scalar G31 6632 G33 G22 G23 G21 G23 G21 G22 detA G11 a12 a13 632 G33 6031 G33 G31 0632 We define the i7j minor of A A deleting row 139 and column j Then Zj as the submatrix of A obtained by det A Q11 det A11 G12 det A12 Q13 det A13 Ex Find the determinant of A Determinant of an n x n Matrix Let A aw be an n gtlt n matrix n 2 2 Then the determinant of A is det A all det A11 G12 det A12 13 det A13 aln det A1 For large matrices this is not practical Luckily some properties of the determinant help out Thm The determinant of a square matrix A is the uniquely defined scalar st the following hold a Adding a multiple of one row to another does not change det A b lnterchanging two rows changes the sign of det A c Multiplying a row by a nonzero scalar multiplies det A by the same d The determinant of an upper lower triangular matrix is equal to the product of its diagonals Using these properties we can find the determinant using another method Thm If PA LU then detA dethet U 1quot U11u22unn where k is the number of row interchanges A is singular iff det A 0 A Determinant Examples Ex Find the determinant of and 1 0 0 1 2 1 1 0 B 1 0 1 0 1 1 1 0 using the direct method and using the theorem Properties of Determinants Prop If A and B are same size square matrices then det det A det B Cor If A is nonsingular then det A l det A Prop If A is square then det AT det A Thm If A is an n gtlt n matrix then det kA kquot det A Note In general there is no formula for det A B A Matlab Examples Consider 1 1 0 A 1 3 2 4 9 5 Matlab has a rank function which estimates the rank of a matrix For small matrices this will always give the exact rank For large matrices with bad values some very small and very large values this will not always work correctly The construct is quite easy as the rank function simply takes a matrix as an argument gtgt A 110132495 rankA Another option to compute the rank would be to find the ref Unfortunately Matlab does NOT compute the ref of a matrix Instead we can compute the rref reduced row echelon form which is similar but contains some reduction of the upper triangular portion of the matrix as well Try A 110132495 rrefA Just like the ref the number of pivots in the rref of A is the same as the rank A Matlab Examples Recall that to solve Ax b in Matlab we would use a command like XAb Let us try this for the previous matrix and b 0 We have A 110132495 b 000 X A b Note we get a warning and the only solution we get is c 0 For homogeneous systems Ab will always give us a zero solution Instead we should do what we would do by hand We reduce Ab to ref Again Matlab does not have an ref function but rref will workjust fine and it will return a simpler result rrefA b The row of zeros in the solution indicates we have rank reduced by 1 so we have 1 free variable We let z t and use backsubstitution to solve for c and y in terms of the free variable t A Matlab Determinants As with most subjects in linear algebra Matlab has a very intuitive way to compute the determinant We simply define a matrix and apply the function det To find the determinant of Al 00000 l ll ll l Axon we do A 31 231 0014 detA Let us say we wish to verify det A10 det A1O We do gtgt A 312310014 detAquot10 detAquot10

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