B=( 1 2
2 1) here B is matrix find B inverse

To find the inverse of a 2x2 matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \), we use the formula:

\[
B^{-1} = \frac{1}{\text{det}(B)} \cdot \text{adj}(B)
\]

where:
- \(\text{det}(B)\) is the determinant of matrix \( B \).
- \(\text{adj}(B)\) is the adjugate of matrix \( B \).

### Step 1: Calculate the Determinant of \( B \)

\[
\text{det}(B) = (1 \times 1) - (2 \times 2) = 1 - 4 = -3
\]

### Step 2: Find the Adjugate of \( B \)

The adjugate of a 2x2 matrix \( B = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[
\text{adj}(B) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For \( B = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \):

\[
\text{adj}(B) = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix}
\]

### Step 3: Compute \( B^{-1} \)

\[
B^{-1} = \frac{1}{-3} \cdot \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} -\frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{1}{3} \end{pmatrix}
\]

Thus, the inverse of matrix \( B \) is:

\[
B^{-1} = \begin{pmatrix} -\frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{1}{3} \end{pmatrix}
\]

Would you like more details or have any questions?

Here are five related questions you might explore:
1. How do you find the determinant of a 3x3 matrix?
2. What is the significance of the determinant being zero when finding the inverse?
3. How can matrix inverses be used to solve systems of linear equations?
4. What happens if you multiply a matrix by its inverse?
5. Can all square matrices be inverted? Why or why not?

**Tip:** Always check the determinant before attempting to find an inverse. If the determinant is zero, the matrix does not have an inverse.

Learn how to find the inverse of matrix B = [1 2; 2 1] with detailed step-by-step calculations using determinant and adjugate. This problem covers key concepts in linear algebra, including matrix operations and inverses. Suitable for advanced high school students.

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