Find the values of 'a' and 'b' for which the following hold :
`[(3,2),(7,a)][(5,-2),(-7,b)]=[(1,0),(0,1)]`.

Answer

To solve the problem, we need to find the values of 'a' and 'b' such that the product of the two matrices equals the identity matrix.

Given matrices:
\[ A = \begin{pmatrix} 3 & 2 \\ 7 & a \end{pmatrix}, \quad B = \begin{pmatrix} 5 & -2 \\ -7 & b \end{pmatrix} \]
We want to find 'a' and 'b' such that:
\[ AB = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 1: Multiply the matrices A and B
To find the product \( AB \), we multiply the rows of matrix A by the columns of matrix B.

The product \( AB \) is calculated as follows:
\[ AB = \begin{pmatrix} 3 & 2 \\ 7 & a \end{pmatrix} \begin{pmatrix} 5 & -2 \\ -7 & b \end{pmatrix} \]

Calculating the elements of the resulting matrix:
- First row, first column:
\[ 3 \cdot 5 + 2 \cdot (-7) = 15 - 14 = 1 \]

- First row, second column:
\[ 3 \cdot (-2) + 2 \cdot b = -6 + 2b \]

- Second row, first column:
\[ 7 \cdot 5 + a \cdot (-7) = 35 - 7a \]

- Second row, second column:
\[ 7 \cdot (-2) + a \cdot b = -14 + ab \]

Thus, we have:
\[ AB = \begin{pmatrix} 1 & -6 + 2b \\ 35 - 7a & -14 + ab \end{pmatrix} \]

### Step 2: Set the product equal to the identity matrix
Now we set the resulting matrix equal to the identity matrix:
\[ \begin{pmatrix} 1 & -6 + 2b \\ 35 - 7a & -14 + ab \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

From this, we can create a system of equations:
1. \( -6 + 2b = 0 \)
2. \( 35 - 7a = 0 \)
3. \( -14 + ab = 1 \)

### Step 3: Solve for 'b' from the first equation
From the first equation:
\[ -6 + 2b = 0 \implies 2b = 6 \implies b = 3 \]

### Step 4: Solve for 'a' from the second equation
From the second equation:
\[ 35 - 7a = 0 \implies 7a = 35 \implies a = 5 \]

### Step 5: Verify with the third equation
Now we substitute \( a = 5 \) and \( b = 3 \) into the third equation to verify:
\[ -14 + ab = 1 \implies -14 + 5 \cdot 3 = -14 + 15 = 1 \]
This holds true.

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 5, \quad b = 3 \]