

Matrix Size & Validation
Vector Algebra What is Vector? Vector Norm Unit Vector Vector Addition Vector Subtraction Vector Scalar Multiple Vector Multiplication Vector Inner Product Vector Outer Product Vector Cross Product Vector Triple Cross Product Vector Triple Dot Product Scalar Triple Product Orthogonal & Orthonormal Vector Cos Angle of Vectors Scalar and Vector Projection Matrix Algebra What is a matrix? Special Matrices Matrix One Null Matrix Matrix Diagonal Is Diagonal Matrix? Identity Matrix Matrix Determinant Matrix Sum Matrix Trace Matrix Basic Operation Is Equal Matrix? Matrix Transpose Matrix Addition Matrix Subtraction Matrix Multiplication Matrix Scalar Multiple Hadamard Product Horizontal Concatenation Vertical Concatenation Elementary Row Operations Matrix RREF Finding inverse using RREF (GaussJordan) Finding Matrix Rank using RREF Matrix Inverse Is Singular Matrix? Linear Transformation Matrix Generalized Inverse Solving System of Linear Equations Linear combination, Span & Basis Vector Linearly Dependent & Linearly Independent Change of basis Matrix Rank Matrix Range Matrix Nullity & Null Space Eigen System Matrix Eigen Value & Eigen Vector Symmetric Matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrix Similarity Transformation and Matrix Diagonalization Matrix Power Orthogonal Matrix Spectral Decomposition Singular Value Decomposition Resources on Linear Algebra 
Matrix Inverse When we are dealing with ordinary number, when we say then we can obtainas long as. We write it asor. For example, the inverse of 2 is ½. Matrix inverse is similar to division operation in ordinary numbers. Suppose we have matrix multiplication such that the result of matrix product is an identity matrix . If such matrix exists, then that matrix is unique and we can write or we can also write. Matrix inverse exists only for a square matrix (that is a matrix that has the same number of rows and columns). Unfortunately, matrix inverse does not always exist. Thus, we give name that a square matrix is singular if that matrix does not have an inverse matrix (remember a single person does not have a spouse). When a square matrix has an inverse, it is called nonsingular matrix. Because matrix inverse is a very important operation, in linear algebra, there are many ways to compute matrix inverse.
Example The interactive program below is using numerical methods. As this is an educational program, I limit the matrix size to square matrix of medium size up to order 10. Random Example button will create new random input matrix. PropertiesSome important properties of matrix inverse are
See also: matrix multiplication, matrix transpose, determinant, rank Rate this tutorial or give your comments about this tutorial Preferable reference for this tutorial is Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\ 



