How to Invert a 2x2 Matrix Using Maxima Computer Algebra System

Understanding Matrix Inversion in Maxima

What We're Solving:

You're trying to find the inverse of a 2×2 diagonal matrix using the Maxima computer algebra system. Specifically, you want to invert the matrix with 4 and 2 on the diagonal and zeros elsewhere.

The Approach:

Matrix inversion is like finding the "opposite" of a matrix - when you multiply a matrix by its inverse, you get the identity matrix (like how 5 × 1/5 = 1). We're using Maxima's built-in functions to both create the matrix and calculate its inverse. This is especially nice because diagonal matrices have a particularly elegant inverse!

Step-by-Step Solution:

Step 1: Create the Matrix ``` matrix([4,0],[0,2]) ```

• This creates a 2×2 matrix where the first row is [4,0] and second row is [0,2]
• The `matrix()` function in Maxima takes each row as a separate list
• This gives us the matrix: $\begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}$

Step 2: Use the Invert Function ``` invert(%) ```

• The `%` symbol in Maxima refers to the last computed result
• So this is equivalent to `invert(matrix([4,0],[0,2]))`
• The `invert()` function calculates the matrix inverse using standard algorithms

Step 3: Understanding What Happens Behind the Scenes For a 2×2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the inverse is $\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

But since our matrix is diagonal (only diagonal elements are non-zero), it's even simpler! For a diagonal matrix, you just take the reciprocal of each diagonal element.

The Answer:

The inverse matrix is: $\begin{pmatrix} \frac{1}{4} & 0 \\ 0 & \frac{1}{2} \end{pmatrix}$

In Maxima, this will display as: ``` matrix([1/4,0],[0,1/2]) ```

Memory Tip:

For diagonal matrices, remember the "flip the diagonals" rule: the inverse of a diagonal matrix is just another diagonal matrix where each diagonal element becomes its reciprocal (1/4 instead of 4, 1/2 instead of 2). The zeros stay zeros! This makes diagonal matrices some of the easiest to invert by hand.

Great job exploring matrix operations with computer algebra systems - they're incredibly powerful tools for checking your work and handling complex calculations! 🎉