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-1 exists (i.e., if A has an inverse). If a matrix does not have an inverse, then it is singular.
Examples Explanation
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Calculating The Inverse
Given 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
Examples
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Explanation
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This system can be expressed as an augmented matrix. Recall that an augmented matrix includes the constant coefficients of a system of equations.
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The solution to this system will provide the elements of M -1.
Solving Systems of Equations with The Matrix Inverse
Systems of linear equations can be solved using the matrix inverse. The system must be represented by the equation
AX=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
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Examples
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Explanation
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The solution is (x,y)=(3,4)
Try these exercises
Solve
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Show that
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Show that the inverse of M is
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Show that
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Find the inverse of N using a system of four equations in four unknowns.
- Is the identity matrix singular or nonsingular?
- Solve the system
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using elementary row operations and matrix inverses.
- Solve the system
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using elementary row orations and matrix inverses.
- Solve the system
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using elementary row operations and matrix inverses.
- Is it possible to solve a system composed of less unknowns than equations using elementary row operations and the matrix inverse?
Answers to questions
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By substitution,
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No elementary row operations are required to transform In into In, so In-1=In. Since In has an inverse, it is nonsingular.
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- Yes, but the extraneous equations must be eliminated so that the coefficient matrix is square (having the same number of equations as unknowns).