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.

Inverses

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

               

Inverses

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

Inverses

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

               

Inverses

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

               

Inverses

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

Inverses

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

Inverses

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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