6
$\begingroup$

Given a matrix $$x = \begin{bmatrix} 40 & 0 & 0 & 0\\ 0 & 80 & 100 & 0 \\ 0 & 40 & 120 & 0 \\ 0 & 0 & 0 & 60\end{bmatrix}$$

How to find the inverse of that matrix? What I know: $\det(x) = ac-bd$, inverse of a 2x2 matrix: $$x^{-1} = \frac{1}{\det(x)}\cdot \begin{bmatrix} d &-b\\ -c &a\end{bmatrix}.$$

There is a lot of content online; however none of them has a specific numerical example.

$\endgroup$

2 Answers 2

11
$\begingroup$

Block diagonal matrices can be inverted block by block. See also [*].

In your example:

$$\begin{bmatrix} 40 & 0 & 0 & 0 \\ 0 & 80 & 100 & 0 \\ 0 & 40 & 120 & 0 \\ 0 & 0 & 0 & 60\end{bmatrix}^{-1} = \begin{bmatrix} [40]^{-1} & \begin{matrix} 0 \quad & 0\quad \end{matrix} & 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{bmatrix} 80 & 100 \\ 40 & 120 \end{bmatrix}^{-1} & \begin{matrix} 0 \\ 0 \end{matrix} \\ 0 & \begin{matrix} 0\quad & 0\quad \end{matrix} & [60]^{-1} \end{bmatrix}. $$

Can you take it from here?

$\endgroup$
2
$\begingroup$

Guide:

If you have a matrix of the form of

$$diag(D_1, D_2, D_3),$$

where each block is invertible, then its inverse is $$diag(D_1^{-1},D_2^{-1}, D_3^{-1}).$$

You should verify this.

In your question $D_2$ is $2$ by $2$ and the other two blocks are scalar.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.