Determine the values of p such that the rank of A = 1 1 -1 0 4 4 -3 1 p 2 2 2 9 9 p 3 is 3

To determine the values of \( p \) such that the rank of the matrix \( A \) is 3, we need to analyze the matrix and ensure that it has three linearly independent rows (or columns). The matrix \( A \) is given by:

\[
A = \begin{pmatrix}
1 & 1 & -1 & 0 \\
4 & 4 & -3 & 1 \\
p & 2 & 2 & 2 \\
9 & 9 & p & 3
\end{pmatrix}
\]

The rank of the matrix is 3 if and only if the determinant of any 3x3 submatrix is non-zero, and at least one 4x4 determinant is zero. Let's start by computing the determinant of the 4x4 matrix \( A \).

### Step 1: Compute the determinant of the 4x4 matrix
\[
\text{det}(A) = \begin{vmatrix}
1 & 1 & -1 & 0 \\
4 & 4 & -3 & 1 \\
p & 2 & 2 & 2 \\
9 & 9 & p & 3
\end{vmatrix}
\]

Expanding this determinant along the first row:

\[
\text{det}(A) = 1 \cdot \begin{vmatrix} 4 & -3 & 1 \\ 2 & 2 & 2 \\ 9 & p & 3 \end{vmatrix} - 1 \cdot \begin{vmatrix} 4 & -3 & 1 \\ p & 2 & 2 \\ 9 & p & 3 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 4 & 4 & 1 \\ p & 2 & 2 \\ 9 & 9 & 3 \end{vmatrix}
\]

### Step 2: Evaluate the 3x3 Determinants
Let’s calculate these three 3x3 determinants.

1. **First determinant:**
\[
D_1 = \begin{vmatrix} 4 & -3 & 1 \\ 2 & 2 & 2 \\ 9 & p & 3 \end{vmatrix}
\]

2. **Second determinant:**
\[
D_2 = \begin{vmatrix} 4 & -3 & 1 \\ p & 2 & 2 \\ 9 & p & 3 \end{vmatrix}
\]

3. **Third determinant:**
\[
D_3 = \begin{vmatrix} 4 & 4 & 1 \\ p & 2 & 2 \\ 9 & 9 & 3 \end{vmatrix}
\]

We calculate each of these determinants and find the condition for them to be non-zero. However, if the 4x4 determinant of \( A \) is non-zero, the rank of \( A \) is 4, and we need it to be 3. Hence, we set the determinant of the full matrix \( A \) to zero and find the corresponding \( p \) values.

Let me compute this step-by-step.import sympy as sp

# Define the matrix A with the variable p
p = sp.symbols('p')
A = sp.Matrix([
    [1, 1, -1, 0],
    [4, 4, -3, 1],
    [p, 2, 2, 2],
    [9, 9, p, 3]
])

# Compute the determinant of the 4x4 matrix A
det_A = A.det()
det_AThe determinant of the 4x4 matrix \( A \) is:

\[
\text{det}(A) = -p^2 - 4p + 12
\]

For the rank of \( A \) to be 3, this determinant must equal zero:

\[
-p^2 - 4p + 12 = 0
\]

### Step 3: Solve for \( p \)
Let's solve the quadratic equation:

\[
p^2 + 4p - 12 = 0
\]

I'll find the roots of this equation to determine the possible values of \( p \).The solutions to the quadratic equation are \( p = -6 \) and \( p = 2 \).

### Conclusion
The values of \( p \) such that the rank of the matrix \( A \) is 3 are \( p = -6 \) and \( p = 2 \).

Would you like further details or explanations? Here are some related questions you might find helpful:

1. What is the geometric interpretation of the rank of a matrix?
2. How can we determine the rank of a matrix without computing the determinant?
3. What happens if \( p \) is not equal to \(-6\) or \(2\)?
4. How is the determinant related to the linear independence of rows or columns?
5. Can the rank of a matrix be higher than the number of rows or columns?

**Tip:** The rank of a matrix provides insight into the dimensionality of the vector space spanned by its rows or columns.

Explore how to determine the values of p such that the rank of matrix A = 1 1 -1 0; 4 4 -3 1; p 2 2 2; 9 9 p 3 is 3. This problem involves calculating determinants, analyzing linear independence of rows, and solving a quadratic equation related to the determinant. Suitable for college-level students.

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