Given `A=[{:(,2,0),(,-1,7):}] and I=[{:(,1,0),(,0,1):}] and A^2 =9A+mI`. Find m.

Answer

To solve the given problem, we need to find the value of \( m \) from the equation \( A^2 = 9A + mI \), where \( A = \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix} \) and \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \).

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

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

\[ A^2 = A \cdot A = \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix} \cdot \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix} \]

Calculating the elements of \( A^2 \):

- First row, first column: \( 2 \cdot 2 + 0 \cdot -1 = 4 \)
- First row, second column: \( 2 \cdot 0 + 0 \cdot 7 = 0 \)
- Second row, first column: \( -1 \cdot 2 + 7 \cdot -1 = -2 - 7 = -9 \)
- Second row, second column: \( -1 \cdot 0 + 7 \cdot 7 = 49 \)

Thus,

\[ A^2 = \begin{pmatrix} 4 & 0 \\ -9 & 49 \end{pmatrix} \]

### Step 2: Calculate \( 9A \)

Next, we calculate \( 9A \):

\[ 9A = 9 \cdot \begin{pmatrix} 2 & 0 \\ -1 & 7 \end{pmatrix} = \begin{pmatrix} 18 & 0 \\ -9 & 63 \end{pmatrix} \]

### Step 3: Substitute into the equation \( A^2 = 9A + mI \)

Now we substitute \( A^2 \) and \( 9A \) into the equation:

\[ A^2 = 9A + mI \]

This gives us:

\[ \begin{pmatrix} 4 & 0 \\ -9 & 49 \end{pmatrix} = \begin{pmatrix} 18 & 0 \\ -9 & 63 \end{pmatrix} + m \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 4: Express \( mI \)

The term \( mI \) can be expressed as:

\[ mI = \begin{pmatrix} m & 0 \\ 0 & m \end{pmatrix} \]

### Step 5: Set up the equation

Now, we can set up the equation by equating the matrices:

\[ \begin{pmatrix} 4 & 0 \\ -9 & 49 \end{pmatrix} = \begin{pmatrix} 18 + m & 0 \\ -9 & 63 + m \end{pmatrix} \]

### Step 6: Solve for \( m \)

From the first row, first column:

\[ 4 = 18 + m \implies m = 4 - 18 = -14 \]

From the second row, second column:

\[ 49 = 63 + m \implies m = 49 - 63 = -14 \]

Both equations give the same value for \( m \).

### Final Answer

Thus, the value of \( m \) is:

\[ \boxed{-14} \]