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### Matrices

 This page is the red pillThe size of a matrix is the number of rows and columns.A matrix with M rows and N columns is a MxN matrix.Individual elements in the matrix is referenced using two index values. The order is row first then column.Common matrices used are 2x2, 3x3 and 4x4.The major diagonal of a matrix is the elements where the row number is equal to the column number. That is Mij where i=j.If all non diagonal elements in a matrix is zero then the matrix is a diagonal matrix.The identity matrix I is a matrix that have a size of NxN and all elements i=j is set to one. All others are zero.TransponseThe transpose of M is known as MT. It is the c*r matrix where the columns are formed from the rows of M. It 'flips' the matrix diagonally. MTij = Mij(MT)T = M. The transpose of a matrix transpose is the matrix again.DT = D for any diagonal matrix.Matrix MultiplyingMultiplying a matrix M with a scalar k ( kM ) is done by multiplying each element in the matrix with the scalar.Multiplying two matricesA r*n matrix A may be multiplied by a n*c matrix B. The result C is a r*c matrix. The number of columns in A needs to be the same as the number of columns in B.Each element Cij will be the sum of the dot product of row i in A with column j in B.Multiplying any matrix M with a square matrix S result in a matrix of the same size as M. If S is the identity matrix I then the result will be M.AB != BAMultiplying a vector and a matrixDeterminantThe determinant of a square matrix M is a scalar denoted |M|.It only exist for square matrices.If any row or column in a matrix is all zero then the determinant of the matrix is zero.||AB|| = |A| |B||MT| = |M|InverseThe inverse of a square matrix M, M-1 is the matrix that when multiplied by M will result in I.Not all matrices have an inverse. A matrix that is invertible have a non zero determinant. MM-1 = I(M-1)-1 = MI-1 = I(MT)-1 = (M-1)TOrthogonalA square matrix is orthogonal if the product of the matrix and it's transpose is the identity matrix. MMT = IIf a matrix is orthogonal then the inverse and the transpose is equal. MT = M-1