If `A=[{:(,-1,1),(,a,b):}] and A^2=I`, find a and b.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) in the matrix \( A \) given that \( A^2 = I \), where \( I \) is the identity matrix. The matrix \( A \) is given as:

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

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} -1 & 1 \\ a & b \end{pmatrix} \cdot \begin{pmatrix} -1 & 1 \\ a & b \end{pmatrix} \]

### Step 2: Perform the matrix multiplication

Now, we will perform the multiplication:

- First row, first column:
\[ (-1)(-1) + (1)(a) = 1 + a \]

- First row, second column:
\[ (-1)(1) + (1)(b) = -1 + b \]

- Second row, first column:
\[ (a)(-1) + (b)(a) = -a + ab \]

- Second row, second column:
\[ (a)(1) + (b)(b) = a + b^2 \]

So, we have:

\[ A^2 = \begin{pmatrix} 1 + a & -1 + b \\ -a + ab & a + b^2 \end{pmatrix} \]

### Step 3: Set \( A^2 \) equal to the identity matrix \( I \)

The identity matrix \( I \) for a 2x2 matrix is:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Setting \( A^2 \) equal to \( I \):

\[ \begin{pmatrix} 1 + a & -1 + b \\ -a + ab & a + b^2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 4: Equate corresponding elements

From the equality of the matrices, we can set up the following equations:

1. \( 1 + a = 1 \)
2. \( -1 + b = 0 \)
3. \( -a + ab = 0 \)
4. \( a + b^2 = 1 \)

### Step 5: Solve the equations

**From equation 1:**
\[ 1 + a = 1 \implies a = 0 \]

**From equation 2:**
\[ -1 + b = 0 \implies b = 1 \]

**Substituting \( a = 0 \) and \( b = 1 \) into equations 3 and 4:**

**Equation 3:**
\[ -a + ab = 0 \implies -0 + 0 \cdot 1 = 0 \quad \text{(True)} \]

**Equation 4:**
\[ a + b^2 = 1 \implies 0 + 1^2 = 1 \quad \text{(True)} \]

### Conclusion

The values of \( a \) and \( b \) are:

\[ a = 0, \quad b = 1 \]