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FutureStarrCalculate Inverse of a Matrixor
In mathematics, the inverse is a type of number that, when multiplied by a number, results in the multiplicand. For example, the inverse of 2 is (−1). The inverse of the result of this compound is also 2.
The inverse of matrix can be found using two methods. The inverse of a matrix can be calculated through elementary operations and through the use of an adjoint of a matrix. The elementary operations on a matrix can be performed through row or column transformations. Also, the inverse of a matrix can be calculated by applying the inverse of matrix formula through the use of the determinant and the adjoint of the matrix. For performing the inverse of the matrix through elementary column operations we use the matrix X and the second matrix B on the right-hand side of the equation.
Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic. (Source: en.wikipedia.org)
Gaussian Elimination is the most useful and easiest way to gain the inverse of matrix, so we should explain it carefully with details and examples. Gaussian Elimination is the way used between each row or column, we can use it the change number of the element in matrix just like the way to solve linear equation with two unknown variables. Then, we use this way to get the identity in the right and the change of identity in the left should be the inverse of that matrix. Take an example of matrix.
Before calculating the inverse of a matrix let us understand what a matrix is? A matrix is a definite collection of objects arranged in rows and columns These objects are called elements of the matrix. The order of a matrix is written as number rows by number of columns. For example, 2 × 2, 2 × 3, 3 × 2, 3 × 3, 4 × 4 and so on. We can find the matrix inverse only for square matrices, whose number of rows and columns are equal such as 2 × 2, 3 × 3, etc. In simple words, inverse matrix is obtained by dividing the adjugate of the given matrix by the determinant of the given matrix. In this article, you will learn what a matrix inverse is, how to find the inverse of a matrix using different methods, properties of inverse matrix and examples in detail. (Source: byjus.com)