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# The Rank of a Matrix and Existence of a Unique Solution to an Equations System

Rank of a Matrix and Existence of a Unique Solution to an Equations System The rank of a matrix is the maximum number of linearly independent rows or columns in it Alternatively it is the order of the largest square sub-matrix possible with a non-zero determinant If the matrix is of order mn then the rank cannot be greater than the smaller of m or n Hence to reduce search time for linear independence choose to investigate either the set of columns or the set of rows depending on whichever dimension is smaller For a square matrix choice doesnt matter for search time Consider the square matrix A whose determinant 0 ie rank 3 as all three columns or rows are linearly independent Now consider the equations system This system has the solution x -33 y -3 and z 19 ie ie ie is a linear combination of the columns of A with the solutions as the weights Suppose that you formed an augmented matrix B by taking the three columns of A and as the fourth column ie B What would be the rank of B ie the maximum number of linearly independent columns in B We hope you answered 3 Can you now see that an equations system Ax b has a solution only if rankA rankAb

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