Let `S` be the set which contains all possible vaues fo `I ,m ,n ,p ,q ,r` for which `A=[I^2-3p0 0m^2-8q r0n^2-15]` be non-singular idempotent matrix. Then the sum of all the elements of the set `S` is ________.

Explanation

To solve the problem, we need to analyze the given matrix \( A \) and determine the conditions under which it is a non-singular idempotent matrix.

Given:
\[ A = \begin{bmatrix} I^2 - 3p & 0 & 0 \\ m^2 - 8q & r & 0 \\ n^2 - 15 & 0 & 0 \end{bmatrix} \]

### Step 1: Understand the properties of idempotent matrices
An idempotent matrix \( A \) satisfies the condition \( A^2 = A \).

### Step 2: Calculate \( A^2 \)
To find \( A^2 \), we multiply \( A \) by itself:
\[ A^2 = A \cdot A = \begin{bmatrix} I^2 - 3p & 0 & 0 \\ m^2 - 8q & r & 0 \\ n^2 - 15 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} I^2 - 3p & 0 & 0 \\ m^2 - 8q & r & 0 \\ n^2 - 15 & 0 & 0 \end{bmatrix} \]

Calculating the elements of \( A^2 \):
- The (1,1) entry: \( (I^2 - 3p)(I^2 - 3p) = (I^2 - 3p)^2 \)
- The (2,2) entry: \( (m^2 - 8q)(r) \)
- The (3,3) entry: \( (n^2 - 15)(0) = 0 \)

Thus, we can write:
\[ A^2 = \begin{bmatrix} (I^2 - 3p)^2 & 0 & 0 \\ (m^2 - 8q)r & r^2 & 0 \\ (n^2 - 15)(0) & 0 & 0 \end{bmatrix} \]

### Step 3: Set the condition for idempotency
Since \( A^2 = A \), we equate the corresponding entries:
1. \( (I^2 - 3p)^2 = I^2 - 3p \)
2. \( (m^2 - 8q)r = m^2 - 8q \)
3. \( r^2 = r \)
4. \( 0 = 0 \) (which is always true)

### Step 4: Analyze the equations
1. From \( (I^2 - 3p)^2 = I^2 - 3p \):
- Let \( x = I^2 - 3p \). The equation becomes \( x^2 = x \), which gives \( x(x - 1) = 0 \). Thus, \( x = 0 \) or \( x = 1 \).
- Therefore, \( I^2 - 3p = 0 \) or \( I^2 - 3p = 1 \).

2. From \( (m^2 - 8q)r = m^2 - 8q \):
- Let \( y = m^2 - 8q \). The equation becomes \( yr = y \), which gives \( y(r - 1) = 0 \). Thus, \( y = 0 \) or \( r = 1 \).
- Therefore, \( m^2 - 8q = 0 \) or \( r = 1 \).

3. From \( r^2 = r \):
- This gives \( r(r - 1) = 0 \), so \( r = 0 \) or \( r = 1 \).

### Step 5: Solve for \( I, m, n, p, q, r \)
1. If \( I^2 - 3p = 0 \):
- \( I^2 = 3p \) implies \( p = \frac{I^2}{3} \).

2. If \( I^2 - 3p = 1 \):
- \( I^2 = 3p + 1 \).

3. For \( m^2 - 8q = 0 \):
- \( q = \frac{m^2}{8} \).

4. For \( n^2 - 15 = 0 \):
- \( n^2 = 15 \) gives \( n = \pm \sqrt{15} \).

### Step 6: Collect possible values
- Possible values for \( I \) from \( I^2 = 3p \) or \( I^2 = 3p + 1 \) yield \( I = \pm 2 \).
- Possible values for \( m \) yield \( m = \pm 3 \).
- Possible values for \( n \) yield \( n = \pm 4 \).
- \( p, q, r \) must be \( 0 \) based on the conditions derived.

### Step 7: Sum all elements of set \( S \)
The possible values are:
- \( I: 2, -2 \)
- \( m: 3, -3 \)
- \( n: 4, -4 \)
- \( p = 0, q = 0, r = 0 \)

Thus, the sum of all possible values:
\[ 2 + (-2) + 3 + (-3) + 4 + (-4) + 0 + 0 + 0 = 0 \]

### Final Answer:
The sum of all the elements of the set \( S \) is \( 0 \).