Question

How to find the inverse of a 3x3 matrix

Answer 1

To find the inverse of a \(3 \times 3\) matrix \(A\), you can use the formula involving the determinant and the matrix of cofactors (adjugate matrix). Here’s a step-by-step guide:

### Step 1: Confirm that the matrix is invertible
First, compute the determinant of \(A\). The matrix \(A\) is invertible if its determinant is non-zero (\(\det(A) \neq 0\)).

For a matrix
\[
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix},
\]
the determinant (\(\det(A)\)) is calculated as:
\[
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).
\]

### Step 2: Calculate the matrix of minors
Next, calculate the minor of each element of the matrix, which is the determinant of the \(2 \times 2\) matrix that remains after removing the row and column of the element.

For example, the minor \(M_{11}\) of element \(a\) is:
\[
M_{11} = \det\begin{pmatrix}
e & f \\
h & i
\end{pmatrix} = ei - fh.
\]

You will do this for all elements \(M_{ij}\) of the matrix.

### Step 3: Calculate the matrix of cofactors
The cofactor \(C_{ij}\) is given by:
\[
C_{ij} = (-1)^{i+j} M_{ij}.
\]
This involves adding a sign based on the position of the element. The cofactor matrix is structured as follows:
\[
C = \begin{pmatrix}
C_{11} & C_{12} & C_{13} \\
C_{21} & C_{22} & C_{23} \\
C_{31} & C_{32} & C_{33}
\end{pmatrix}.
\]

### Step 4: Calculate the adjugate (adjoint) matrix
The adjugate matrix is the transpose of the cofactor matrix \(C\):
\[
\text{adj}(A) = C^T.
\]

### Step 5: Calculate the inverse using the formula
Finally, the inverse of the matrix \(A\) can be computed using the formula:
\[
A^{-1} = \frac{1}{\det(A)} \text{adj}(A).
\]

### Example
Let’s find the inverse of the matrix:
\[
A = \begin{pmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{pmatrix}.
\]

1. Calculate the determinant:
\(\det(A) = 1(1 \cdot 0 - 4 \cdot 6) - 2(0 \cdot 0 - 4 \cdot 5) + 3(0 \cdot 6 - 1 \cdot 5) = -24 + 40 - 15 = 1.\)

2. Calculate minors and cofactors:
- \(M_{11} = -24, C_{11} = -24\)
- \(M_{12} = -15, C_{12} = 15\)
- \(M_{13} = 6, C_{13} = 6\)
- and so on for all elements.

3. Assemble the cofactors into the cofactor matrix, then transpose it to get the adjugate.

4. Finally:
\[
A^{-1} = \frac{1}{1} \text{adj}(A) = \text{adj}(A).
\]

This gives you the inverse matrix.

By following these steps carefully, you can compute the inverse of any \(3 \times 3\) matrix as long as the determinant is not zero.