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Math for Quantitative Finance

# Inverses

The main application of inverse matrices is in solving matrix equations. For example, if you know that $$AB=C$$, and you only know the matrices $$B$$ and $$C$$, you need to be able to find $$A.$$ These kinds of equations arise when fitting a line to a set of data, and being able to solve them is essential for many applications.

Let $A=\left[\begin{array}{cc} 1&0\\ -1&4\end{array}\right].$ What is the inverse of $$A?$$

The inverse is often combined with matrix operations, but the resulting matrix is not always what you'd expect.

If $$A$$ and $$B$$ are square matrices, which of the matrices below are equal to $$AB$$? $\text{I. } \left(A^{-1}B^{-1}\right)^{-1}\qquad \text{II. }\left(B^{-1}A^{-1}\right)^{-1}.$

Which of the following is always equal to $$\left(A^T\right)^{-1}$$?

Unfortunately, matrices do not always have inverses. In these situations then, it is no longer possible to solve matrix equations.

For what value of $$a$$ does the matrix $\left[\begin{array}{cc} 9&a\\ 12&4\end{array}\right]$ have no inverse?

If $$A$$ and $$B$$ are invertible matrices, and so is $$A+B,$$ then what is the inverse of $$A^{-1}+B^{-1}?$$

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