Is there a name for matrices that have this property?  A_ij = 1/A_ji? (i.e. symmetric with inverse)?
599592
At first I thought you were wanting the orthogonal matrices, but your example shows you are not. Orthogonal matrices are those whose inverse is their transpose. Yours are ones where the elements of the transpose are the inverse of the elements of the original.\n\nCertainly such a matrix has to have +-1 on the diagonal. Your matrix is singluar and any 2x2 matrix with your property and all 1 on the diagonal will be also. This means that one eigenvalue is 0 and the other is 2.\n\nI don't see anything special for 3x3 offhand, but I will look further.
332712
The statement is true for :\nIf A is a skew-symmetric matrix, then A = - At (A transpose)\nSo (At)^(-1)= (A^(-1))t\nSo (-A)^(-1)= (A^(-1))t\nSo -1((A)^(-1)= (A^(-1))t\nSo A^(-1) is skew-symmetric
609321
One is the inverse of the other
909215
It can be encoded with another 1*2 matrix through addition.\n\nFor example,\n\noriginal matrix = (x   y)\n\nencoding matrix = (a b)\n\nCoded matrix = (x+a    y+b)
718910
ill use this as a key for a standard matrix:\n\na b\nc d\n\nin your coeff. matrix:\n\na=equation 1 x coeff\nb=eq.1 y coeff.\nc=eq.2 x coeff.\nd=eq.2 y coeff.\n\nso the coeff. matrix looks like\n\n3   11\n-1  4\n\nthe value of the determinant is always "ad-bc" for a 2 by 2, so yours youls be\n\n(3)(4)-(-1)(11) = 12+11 = 23\n\n\nand i dont know what you mean by adjoint, sorry
813865
The singular value decomposition for a matrix A writes A as a product (hanger)(stretcher)(aligner). \n \n\nIt's an amazing and useful fact that every m x n matrix has a singular value decomposition. \n\n-------------------\nIn linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics.\n\nThe SVD can be seen as a generalization of the spectral theorem to arbitrary, not necessarily square, matrices. The spectral theorem says that normal matrices can be unitarily diagonalized using a basis of eigenvectors.\n\n---------------------------------\nSingular value decomposition takes a rectangular matrix of gene expression data (defined as A, where A is a n x p matrix) in which the n rows represents the genes, and the p columns represents the experimental conditions.\n\n---------------------------------\nA very powerful set of techniques dealing with sets of equations or matrices that are either singular or numerically very close to singular is the so-called singular value decomposition (SVD). SVD allows one to diagnose the problems in a given matrix and provides numerical answer as well.\n\nAny m x n matrix a (m >= n) can be written as the product of an m x n column-orthogonal matrix u, an n x n diagonal matrix with positive or zero elements, and the transpose of an n x n orthogonal matrix v:\n\n\nI hope that all these are useful, good luck!!!!!!!!!!!!
326332
If A is a 2x2 matrix, then 2A means to multiply each of the numbers \n[12\n43]\nby 2 (this is called scalar multiplication)\nand you would get\n[24\n86]\n\nI hope this helps.\n:o)
441742
Assume there is a pair a_i, b_j with a non zero product.\nWrite fg= (a_i x^i + other terms) (b_j x^j + other terms) =\na_i b_j x^(i+j) is non zero.\n\nContradiction.
802863
| A - I(lamda)| = 0 ==> eigenvalue ... \n\nlamda is an eigenvalue...\n\nsolve the matrix first
856009
false,\nuse the identity and -identity,\nboth are invertible, but the sum is the zero matrix, that is NOT invertible.....
1111281
AB is only defined if A has the same number of columns as B has rows.  So two matrices with different dimensions can be multiplied if that constraint holds.
469735
The system of equations:\n[a11 a12 a13; a21 a22 a23]\n[1 -3 4; 2 3 1]\n\nPerforming indicated row operation:\n[a11 a12 a13; a21' a22' a23']\n[1 -3 4; -1 -6 3]\n\na21' = a11 - a21 = 1 - 2 = -1\na22' = a12 - a22 = -3 - 3 = -6\na23' = a13 - a23 = 4 - 1 = 3
98990
A system of partial differential equations can be described as an nxn matrix (lets call it A). The eigenvector (X) will be the n dimensional vector which has the property that there exists some real eigenvalue (L) such that AX=LX.
470202
Rank is basically the number of nonzero rows in a row-echelon reduced matrix.  So if you have a square matrix with one row of zeroes, you may be looking at something like this:\n\n| 1 0 0 |\n| 0 1 4 |\n| 0 0 0 |\n\nadj(A) is found by finding the matrix made up of the cofactors of A, and then transposing it, and changing signs in a checkerboard pattern in the matrix.  Creating the cofactor matrix is easy - for the new matrix C, Cij is found by finding the determinant of the matrix that results if you elimiate row i, column j in matrix A.  The cofactor matrix for the above example would be\n\n| 0 0 0 |\n| 0 0 0 |\n| 0 4 1 |\n\nand its transpose would be\n\n| 0 0 0 |\n| 0 0 -4 |\n| 0 0 1 |\n\nwhich is a matrix of rank 1.\n\nThe row of zeros creates multiple rows of zeros when creating the cofactor matrix.  That makes sense - when finding the determinant of a matrix with a row of zeros, the determinant always equals 0.  It turns out that the only time that you create a nonzero row in your cofactor matrix is when you eliminate the single row of zeros.
684363
or any matrix with all zero and one 1 at any place on the diagonal\n\n\nI think there is no INVERTIBLE matrix beside the identity matrix that has AA=A. I read today that by saying a matrix is invertible, u r equivalently saying that its kernel is 0.\n\nker(A)=0 means Ax=0 only when x=0. With x replaced by A-I we have A-I=0, hence, A=I.
474193
There are many general properties of determinants of matrices for example\n\nConsider two matrices A and B then\ndet(AB)= det(A) det(B)\n\ndet(A) = det( Transpose(A) )\n\nIf you interchange two rows in a matrix then you will change the sign of the determinant.  Interchanging two columns does the same thing.\n\nThere are many more but these should get you started
542644
Here are a few websites which explain it fairly well...\n\nhttp://www.mai.liu.se/~halun/matrix/\n\nhttp://www.easycalculation.com/matrix/matrix-multiplication.php\n\nThat one has a matrix calculator online. Good luck!