Computing the Inverse of a Matrix For the sake of convenience, denote A-¹ as simply X. Then AX = I. If e, denotes the co

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Computing The Inverse Of A Matrix For The Sake Of Convenience Denote A As Simply X Then Ax I If E Denotes The Co 1 (117.78 KiB) Viewed 79 times

Computing the Inverse of a Matrix For the sake of convenience, denote A-¹ as simply X. Then AX = I. If e, denotes the column i of the identity matrix, then column i of the inverse matrix, call it X(), is the solution of the system of equations: AX (₁) = ei Which can be found using Gaussian elimination. Observing that the required row operations to find the solution of a system of equations do not depend on the right hand side of the equations, the computation of the inverse can be performed by forming the augmented matrix [A | I], and then doing row operations to obtain [I | X]. The matrix, X, on the right is the inverse of A. The condition for the inverse to exist is that the reduced row echelon form of A be the identity matrix. This is not necessarily the most convenient condition to check, and we'll have more to say about this later. The R code Inverse (as usual in the Files folder on Canvas) implements the method for finding the inverse of a square matrix described above. The code computes AA-¹ as a check, and also computes the inverse using the built-in R function solve: an apt name for the function, since when A invertible the unique solution of Az = b is x = A-¹b Exercise 26. Let A be the matrix 20 01 1 A-¹ = -1 1 Using Gaussian elimination, as described above, show that 1 -1 2 -1 2 -2 Exercise 27. Suppose A and B are invertible nxn matrices, show that the inverse of AB is B-¹A-¹. Exercise 28. Suppose A is invertible. Show that (A)−¹ = (A−¹)'. Exercise 29. Suppose A is a concrete 3 x 3 invertible matrix, and B is a concrete 3× 5 matrix. How does one efficiently solve AX B? (Of course, the solution matrix X must be a 3 × 5 matrix.)