To invert a 2x2 matrix, the formula for the inverse is:

\[
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

The inverse of matrix \( A \), denoted \( A^{-1} \), is given by:

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For the matrix:

\[
A = \begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}
\]

We identify:

- \( a = 3 \)
- \( b = 1 \)
- \( c = -1 \)
- \( d = 1 \)

Now, calculate the determinant \( ad - bc \):

\[
\text{Determinant} = (3)(1) - (1)(-1) = 3 + 1 = 4
\]

Since the determinant is non-zero, the inverse exists. Now, apply the formula for the inverse:

\[
A^{-1} = \frac{1}{4} \begin{pmatrix} 1 & -1 \\ 1 & 3 \end{pmatrix}
\]

So,

\[
A^{-1} = \begin{pmatrix} \frac{1}{4} & \frac{-1}{4} \\ \frac{1}{4} & \frac{3}{4} \end{pmatrix}
\]

This is the inverse of the matrix.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How do you find the inverse of a 3x3 matrix?
2. What is the geometric interpretation of matrix inversion?
3. How do you calculate the determinant of larger matrices?
4. What happens if the determinant is zero?
5. How can matrix inversion be used in solving systems of linear equations?

Tip: Always check that the determinant is non-zero before attempting to invert a matrix!

Learn how to calculate the inverse of a 2x2 matrix with the example 3 1, -1 1. This detailed explanation covers the determinant calculation and the formula for matrix inversion. Ideal for Grades 10-12 students studying linear algebra.

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