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Topic: Inverse matrix

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Matrix (mathematics)

Matrix inversion

Normal matrix

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Symmetric matrix

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	PlanetMath: matrix inverse
		Some caveats: computing the matrix inverse for ill-conditioned matrices is error-prone; special care must be taken and there are sometimes special algorithms to calculate the inverse of certain classes of matrices (for example, Hilbert matrices).
		The need to find the matrix inverse depends on the situation- whether done by hand or by computer, and whether the matrix is simply a part of some equation or expression or not.
		However, the matrix inverse may exist in the case of the elements being members of a commutative ring, provided that the determinant of the matrix is a unit in the ring.
	planetmath.org /encyclopedia/Inverse2.html (464 words)

	6.3 - The Inverse of a Square Matrix
		We knew that for a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero.
		The inverse of a square matrix A, denoted by A
		Row-reduce (I suggest using pivoting) the matrix until the left side is the Identity matrix.
	www.richland.edu /james/lecture/m116/matrices/inverses.html (1467 words)

	A 4x4 Matrix Inverse
		In linear algebra, a block LU decomposition is a decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula.
		Once the cofactor matrix is computed, the result is used to calculate the determinant and also the adjoint matrix.
		The adjoint of A is the transpose of the matrix of cofactors and is denoted by adj(A).
	www.cellperformance.com /articles/2006/06/a_4x4_matrix_inverse_1.html (3286 words)

	Online Math Calculators and Solvers
		Graphing Calculator For Inverse Functions An online graphing calculator to draw the graph (in red) formed by reversing the ordered pairs corresponding to all points on the graph (blue) of a function f that you input.
		Determinant of a 2 by 2 Matrix - Calculator.
		Determinant of a 3 by 3 Matrix - Calculator.
	www.analyzemath.com /Calculators.html (989 words)

	The Matrix Inverse
		We can use the previously developed Gauss-Jordan subroutine to find the inverse of a matrix.
		The entries in the matrix for this exercise must be typed in by hand in order to make sure that only six decimal places are stored in the computer.
		Since we require the computation to be as exact as possible and an exact formula is known for the inverse of the Hilbert matrix, this is an example where an inverse matrix comes in handy.
	math.fullerton.edu /mathews/n2003/InverseMatrixMod.html (214 words)

	Matrix Inverse Calculator 1.0 - Uses a complicated method to find the inverse of any square matrix
		You simply choose your matrix's dimensions, and then enter the elements of the matrix you want inverted in the left frame.
		When you press Enter, the inverse matrix will be shown in the right frame.
		If your matrix doesn't have an inverse, you will be told so.
	www.downloadthat.com /windows/Educational/Mathematics/Matrix-Inverse-Calculator.html (254 words)

	Matrix Inverse Example Calutation from determinant, adjoint Tutorial, formula.
		Matrix Inverse Example Calutation from determinant, adjoint Tutorial, formula.
		After calculating determinant, adjoint from the matrix as in the previous tutorials a) Find determinant of A (A)
		c)Forming Minors Matrix of A: Matrix of minors
	www.easycalculation.com /matrix/inverse-matrix-tutorial.php (191 words)

	Review from MAA Monthly of Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
		The book is cleverly structured, and allows one to omit the most difficult topics and yet have a coherent text on a more standard level, so that it can be used successfully with students on several levels.
		Many theorems throughout the book are stated in terms of the length of a matrix in order to simplify the verification of hypotheses.
		The inverse function theorem is carefully stated so that if the conditions of the theorem are satisfied, then Kantorovich's theorem applies, and Newton's method can be used to find the inverse function.
	matrixeditions.com /VC.MAAreview.html (1760 words)

	Matrix and Quaternion FAQ (Site not responding. Last check: )
		For all powers, the matrix must be square, that is orthogonal and the same width and height For example, -1 M is the inverse of the matrix 0 M generates the identity matrix 1 M leaves the matrix unchanged.
		Calculating the inverse of a matrix is a task often performed in order to implement inverse kinematics using spline curves.
		The inverse of an identity matrix is the identity matrix.
	www.j3d.org /matrix_faq/matrfaq_latest.html (7736 words)

	Matrix
		The elements of the matrix are denoted by a
		One of the most useful properties of the matrix is the matrix multiplication.
		= the inverse matrix of a, and I = the identity matrix.
	kwon3d.com /theory/vect/matrix.html (456 words)

	Matrix (mathematics) Summary
		In an identity matrix, all elements are 0 except the diagonal elements that are all equal to 1.
		The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
		The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
	www.bookrags.com /Matrix_(mathematics) (3268 words)

	Invertible matrix Summary
		That is a matrix with all of its (i,j) entries equal to zero whenever i > j.
		Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
		Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices).
	www.bookrags.com /Invertible_matrix (1462 words)

	Matrix Inverses via Technology: Good News and Bad News
		Prerequisites: Students should be familiar with matrix multiplication, elementary row operations that lead to a reduced row echelon form, augmented matrices, and the definition of inverse of a matrix.
		To find the inverse of a matrix A, the students quickly learn to augment A with an appropriately sized identity matrix, rref the augmented matrix, and then examine the right side of the reduced row echelon form of the augmented matrix to immediately obtain the matrix inverse.
		At this point, armed with their new tool for easily finding matrix inverses, the students are ready to end class.
	mathdemos.gcsu.edu /mathdemos/matrix_inverse1 (783 words)

	PMtrxDouble::Inverse
		The Inverse member function returns as its functional result a pointer to a newly-obtained PMtrxDouble object encapsulating the inverse of the matrix encapsulated by the presenting PMtrxDouble object, provided such an inverse matrix can be computed and so encapsulated; otherwise, a null pointer is returned.
		An inverse matrix is considered to be that matrix which, when multiplied by the original matrix, yields an identity matrix (that is a diagonal matrix of unit values).
		Meanwhile, the inverse result matrix enumeration in increment in the C manner, that is with the right-most index varying most rapidly.
	www.lerc.nasa.gov /WWW/price000/pfc/htm/pmtrxdouble_inverse.html (482 words)

	MATRIX ARTICLE
		Calculating the inverse of a matrix is a task often performed in order to implement inverse kinematics using spline curves.
		The inverse of an identity matrix is the identity matrix.
		In this case, the value of each matrix element is defined by: -1 1 i+j M = ----- * det submat M * -1 j,i det M i,j Q22.
	www.morrowland.com /apron/article/technical/matrix/index.php (7421 words)

	News \| TimesDaily.com \| TimesDaily \| Florence, AL (Site not responding. Last check: )
		In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
		Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices).
		For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations; however, for a unique solution, it is necessary that the matrix involved be invertible.
	www.timesdaily.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Inverse_matrix (893 words)

	Linear Algebra (Math 2318) - Systems of Equations and Matrices - Inverse Matrices and Elementary Matrices (Site not responding. Last check: )
		Before proceeding we need to show that the inverse of a matrix is unique, that is for a given invertible matrix A there is exactly one inverse for the matrix.
		As the definition of an inverse matrix suggests, not every matrix will have an inverse. Here is an example of a matrix without an inverse.
		matrix and we’ve shown that the second row of BC is all zeroes and hence the product will never be the identity matrix and so B can’t have an inverse and so is a singular matrix.
	tutorial.math.lamar.edu /AllBrowsers/2318/InverseMatrices.asp (1022 words)

	Matrix Math
		Because of the above (matrix 1 width = matrix 2 height) transformation matrices are therefore square, - 3x3 or 4x4 are commonly used for 3D transformations.
		The inverse of a matrix is a matrix of the same size, which undoes any transformation performed by the original matrix.
		You calculate the determinant of the submatrix, divide that by the determinant of the matrix (not the submatrix), and multiply by
	easyweb.easynet.co.uk /~mrmeanie/matrix/matrices.htm (884 words)

	Sec 4.4: Identity and Inverse Matrices
		For square matrix [A] the inverse is a matrix of same dimension having 1’s along the diagonal and 0’s everywhere else.
		Inverses are used to solve equations by eliminating unwanted values.
		Using matrices to code (scramble) a sensitive message is a primary purpose for matrix multiplication (and for students, it is also fun).
	www.geocities.com /m_felz/Honorsnotes/4.4notes.htm (177 words)

	System of Linear Equations and Matrix Inversion
		The coefficient matrix must be a squared-matrix appearing on the upper left corner of the Data-Matrix, therefore, do not leave any blank rows in between.
		To find the inverse of a square matrix of size n, solve n systems of equations with a unit vector as their right hand side.
		Notice that the coefficient of the variables X1 and X2 are matrix A in both systems of equations, however the RHS are two identity vectors in n=2 dimensional space.
	home.ubalt.edu /ntsbarsh/Business-stat/otherapplets/SysEq.htm (690 words)

	Identity and Inverse Matrices
		The identity matrix I for multiplication is a square matrix with a 1 for every element of the principal diagonal (top left to bottom right) and a 0 in all other positions.
		If an identity matrix is the answer to a problem under matrix multiplication, then each of the two matrices is an inverse matrix of the other.
		The matrix which when multiplied by the original matrix gives the identity matrix as the solution.
	www.blc.edu /fac/rbuelow/Cal/nt6-4.htm (209 words)

	Matrix Inverse Calculator 100% Clean Program - Matrix Inverse Calculator Antivirus Report at datapicks.com (Site not responding. Last check: )
		Matrix Inverse Calculator software have been tested by datapicks.com team against viruses, spyware, adware and was founded to be
		You simply choose your matrix's dimensions, and then enter the elements of the matrix you want inverted in the left frame.
		When you press Enter, the inverse matrix will be shown in the right frame.
	matrix-inverse-calculator.antivirusreport-458-35095.datapicks.com (212 words)

	The inverse of an nxn matrix
		Briefly, matrix inverses behave as reciprocals do for real numbers : the product of a matrix and it's inverse is an identity matrix.
		Augment the nxn matrix A with the nxn identity matrix
		Below is the same matrix A, augmented by the 3x3 identity matrix.
	math.uww.edu /faculty/mcfarlat/inverse.htm (441 words)

	Inverse Matrix Applications
		where the unit matrix has diagonal elements equal to unity, that is equal to 1, and all other non-diagonal matrix elements equal to 0.
		So for a square matrix of coefficients A the inverse matrix A^-1 is the solution of the "n" equations in "n" unknowns.
		Vince Calder ==================================================================== S.K., One significant use of an inverse matrix is in mathematical problem-solving.
	www.newton.dep.anl.gov /askasci/math99/math99249.htm (214 words)

	Maths - Matrix algebra - Inverse - Martin Baker
		The inverse of a 2x2 matrix is shown here.
		Each element of the inverse is the determinant of its minor divided by the determinant of the whole.
		The inverse of a translation by (tx,ty,tz) is a translation by (-tx,-ty,-tz) just move it back in the opposite direction, The translate transform is often represented by a 4x4 matrix together with the multiplication operator as described here.
	www.euclideanspace.com /maths/algebra/matrix/functions/inverse/index.htm (640 words)

	Math Forum - Ask Dr. Math
		You may know that when you multiply any n-by-n matrix by the identity matrix (the n-by-n matrix that has all ones down the main diagonal and zeroes everywhere else), you get that same matrix back again.
		Well, the inverse of an n-by-n matrix, A, is the matrix you can multiply A by to get I, just like the inverse (reciprocal) of a real number, x is the number you can multiply x by (1/x) to get 1.
		You can check this by multiplying it by the original matrix using the method taught to you by Ken. There is another method for finding inverses involving things called determinants.
	mathforum.org /library/drmath/view/55464.html (804 words)

	GaxParaMtxDoub::Inverse
		Note that, currently, the inverse matrix is defined only for matricies that can be demonstrated to be two-dimensional, square, and of a size greater than or equal to two.
		A PString string object to encapsulate the expected weak reference to the obtained inverse matrix instance must either be identified or a new one must be obtained.
		A weak reference to the GMtxDoub double matrix instance encapsulating the inverse matrix is obtained and relicated in the identified PString string object.
	www.grc.nasa.gov /WWW/price000/pac/htc/gaxparamtxdoub_inverse.html (311 words)

	tutOR: Matrix Inverse
		This centre was established to provide help to visitors who are not familiar with the ideas of the inverse of a matrix and/or with a method of finding an inverse using row operations.
		We then use row operations to try to reduce the matrix so that we get the identity on the left where we originally had A. If you don't know about row operations, read the material on Row Operations.
		There are other simple ways of showing whether a matrix has an inverse, see for example the discussion on Determinants.
	www.tutor.ms.unimelb.edu.au /matrix/matrix_inverse.html (775 words)

	Inverse Method
		A Matrix object representing the inverse of the current matrix.
		The following example demonstrates both the Inverse method and the use of the LU class to compute a matrix inverse.
		An alternative method for computing the matrix inverse is to use the LU object.
	www.bluebit.gr /net/Library/Matrix-Inverse.html (193 words)

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