The Inverse of a Matrix University of Michigan(密歇根大学的逆矩阵).pdf
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The Inverse of a Matrix
Francis J. Narcowich
Department of Mathematics
Texas AM University
January 2006
1 Definition of the Inverse
Inverses are defined only for square matrices. Thus, we start with an n ×n (square)
matrix A. We say that an n × n matrix B is an inverse for A if and only if
AB = BA = I , where I is the n × n identity matrix.
The reason that we want to consider inverses for matrices is that they enable
us to easily obtain solutions to linear systems of equations. If we want to solve
Ax = b, where x, b are in Rn or Cn (that is, they are real or complex n × 1
columns), and if B is an inverse for A, then consider this chain of equations:
B (Ax) = Bb
(BA )x = Bb
I
Ix = Bb
x = Bb
The point is that if we know an inverse B for A, then the solution to Ax = b is
just x = Bb.
An example of a matrix that has an inverse is A = 1 2 . It is easy to
1 3
check that 3 −2 is inverse to A. Simply observe that
−1 1
1 2 3 −2 = 3 −2 1 2 = 1 0 .
1 3 −1 1 −1 1 1 3 0 1
There are two important questions that we want to answer. We will start with
the easier of the two. If a matrix A has an inverse B, can it have another inverse
C = B ?
1
The answer is no. Let B and C be inverses for A. Then, AB = I and CA = I .
Thus, C = CI = C (AB) = (CA)B = IB
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