The Inverse of A Matrix |
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Let A be an n*n matrix. An n*n matrix A-1 such that AA-1=A-1A=In is the inverse of A. A matrix A is nonsingular if A-1exists (i.e., if A has an inverse). If a matrix does not have an inverse, then it is singular.
Calculating The InverseGiven a square matrix M, we know the size of its inverse (the same size as M) and the product of M and its inverse. Using this information, the inverse of M can be calculated by assigning variables to the elements of M -1 and representing the product MM -1 as a system of n equations in n unknowns, where M has dimension n*n.
Solving Systems of Equations with The Matrix InverseSystems of linear equations can be solved using the matrix inverse. The system must be represented by the equationAX=B where A is the coefficient matrix of the system, X is the column matrix of variables, and B is the column matrix of constant coefficients. The solution to the system will then be A -1B because
Try these exercises
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1. | Show that |
2. | Show that the inverse of M is |
3. | Show that |
4. | Find the inverse of N using a system of four equations in four unknowns. |
5. | Find the inverse of N using a system of four equations in four unknowns. |
6. | Is the identity matrix singular or nonsingular? |
7. | Solve the system using elementary row operations and matrix inverses. |
8. | Solve the system using elementary row orations and matrix inverses. |
9. | Solve the system using elementary row operations and matrix inverses. |
10. | Is it possible to solve a system composed of less unknowns than equations using elementary row operations and the matrix inverse? |
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