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…, such that
Solving for A yields
or, equivalently,
(5)
This equation expresses A as a product of elementary matrices (since the inverse of an elementary
matrix is also elementary by Theorem 1.5.2). The result now follows from Example 3.
EXAMPLE 5 Geometric Effect of Multiplication by a Matrix
Assuming that and are positive, express the diagonal matrix
as a product of elementary matrices, and describe the geometric effect of multiplication by A in terms of compressions and
expansions.
Solution
From Example 1 we have
which shows that multiplication by A has the geometric effect of compressing or expanding by a factor of in the x-direction and
then compressing or expanding by a factor of in the y-direction.
EXAMPLE 6 Analyzing the Geometric Effect of a Matrix Operator
Express
as a product of elementary matrices, and then describe the geometric effect of multiplication by A in terms of shears, compressions,
expansions, and reflections.
Solution
A can be reduced to I as follows:
