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Matrix Inverse (QF)


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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