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Consider the product
The entry in the th row and th column of the product is
(see the shaded lines above). (8)
, then the 's and the
If , then 8 is the cofactor expansion of along the th row of (Theorem 2.1.1), and if
cofactors come from different rows of , so the value of 8 is zero. Therefore,
(9)
Since is invertible, . Therefore, Equation 9 can be rewritten as
Multiplying both sides on the left by yields
EXAMPLE 7 Using the Adjoint to Find an Inverse Matrix
Use 7 to find the inverse of the matrix in Example 6.
Solution . Thus
The reader can check that
Applications of Formula 7
Although the method in the preceding example is reasonable for inverting matrices by hand, the inversion algorithm
discussed in Section 1.5 is more efficient for larger matrices. It should be kept in mind, however, that the method of Section
1.5 is just a computational procedure, whereas Formula 7 is an actual formula for the inverse. As we shall now see, this
formula is useful for deriving properties of the inverse.
In Section 1.7 we stated two results about inverses without proof.
