Page 151 - 35Linear Algebra

7.5 Inverse Matrix 151

Figure 7.1: The formula for the inverse of a 2×2 matrix is worth memorizing!

Thus, much like the transpose, taking the inverse of a product reverses
the order of the product.

T
T
T
−1
T
3. Finally, recall that (AB) = B A . Since I T = I, then (A A) =
−1 T
T
−1 T
T
−1 T
A (A ) = I. Similarly, (AA ) = (A ) A = I. Then:
−1 T
T −1
(A ) = (A )
2 × 2 Example
7.5.2 Finding Inverses (Redux)
Gaussian elimination can be used to find inverse matrices. This concept is
covered in chapter 2, section 2.3.2, but is presented here again as review in
more sophisticated terms.
Suppose M is a square invertible matrix and MX = V is a linear system.
The solution must be unique because it can be found by multiplying the
equation on both sides by M −1 yielding X = M −1 V . Thus, the reduced row
echelon form of the linear system has an identity matrix on the left:

M V ∼ I M −1 V
Solving the linear system MX = V then tells us what M −1 V is.

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