If `A=[{:(,a,0),(,0,2):}], B=[{:(,0,-b),(,1,0):}], M=[{:(,1,-1),(,1,1):}]` and `BA=M^2`, find the values of a and b.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the matrices \( A \), \( B \), and \( M \) and the equation \( BA = M^2 \).

### Step 1: Define the matrices
Given:
- \( A = \begin{pmatrix} a & 0 \\ 0 & 2 \end{pmatrix} \)
- \( B = \begin{pmatrix} 0 & -b \\ 1 & 0 \end{pmatrix} \)
- \( M = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \)

### Step 2: Calculate \( M^2 \)
To find \( M^2 \), we multiply \( M \) by itself:
\[ M^2 = M \times M = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 1 \cdot 1 + (-1) \cdot 1 = 1 - 1 = 0 \)
- First row, second column: \( 1 \cdot (-1) + (-1) \cdot 1 = -1 - 1 = -2 \)
- Second row, first column: \( 1 \cdot 1 + 1 \cdot 1 = 1 + 1 = 2 \)
- Second row, second column: \( 1 \cdot (-1) + 1 \cdot 1 = -1 + 1 = 0 \)

Thus,
\[ M^2 = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix} \]

### Step 3: Calculate \( BA \)
Now, we calculate \( BA \):
\[ BA = B \times A = \begin{pmatrix} 0 & -b \\ 1 & 0 \end{pmatrix} \times \begin{pmatrix} a & 0 \\ 0 & 2 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 0 \cdot a + (-b) \cdot 0 = 0 \)
- First row, second column: \( 0 \cdot 0 + (-b) \cdot 2 = -2b \)
- Second row, first column: \( 1 \cdot a + 0 \cdot 0 = a \)
- Second row, second column: \( 1 \cdot 0 + 0 \cdot 2 = 0 \)

Thus,
\[ BA = \begin{pmatrix} 0 & -2b \\ a & 0 \end{pmatrix} \]

### Step 4: Set \( BA \) equal to \( M^2 \)
Now we set \( BA \) equal to \( M^2 \):
\[ \begin{pmatrix} 0 & -2b \\ a & 0 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix} \]

### Step 5: Solve the equations
From the equality of the matrices, we can equate corresponding elements:
1. From the first row, first column: \( 0 = 0 \) (no information)
2. From the first row, second column: \( -2b = -2 \)
- Solving gives \( b = 1 \)
3. From the second row, first column: \( a = 2 \)
4. From the second row, second column: \( 0 = 0 \) (no information)

### Conclusion
The values are:
- \( a = 2 \)
- \( b = 1 \)