For what values of K the matrix [4 4 -3 1
            1 1 -1 0
           K 2 2 2
            9 9 K 3]  has rank 3

To determine the values of \( K \) such that the matrix has a rank of 3, we need to find when the determinant of the 4x4 matrix is zero. For a 4x4 matrix to have a rank of 3, its determinant must be zero, but there should be at least one 3x3 submatrix with a non-zero determinant.

The matrix is:

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

### Step-by-Step Process:

1. **Find the determinant of the 4x4 matrix**:
   We compute the determinant of this 4x4 matrix. For this matrix to have rank 3, the determinant of the 4x4 matrix must be zero.

2. **Compute minor determinants**:
   After ensuring the 4x4 determinant is zero, we need to check the determinant of some 3x3 submatrices to ensure that at least one 3x3 submatrix has a non-zero determinant.

Let's calculate this determinant and analyze the results.The determinant of the matrix is:

\[
K^2 + 4K - 12
\]

For the matrix to have a rank of 3, this determinant must be zero. Therefore, we solve:

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

Let's solve this quadratic equation for \( K \).It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can solve the quadratic equation \( K^2 + 4K - 12 = 0 \) manually or with a calculator to find the values of \( K \).

Would you like help with solving it step by step?

Here are some related questions that might interest you:
1. What is the rank of a matrix, and how is it calculated?
2. What happens when the determinant of a matrix is zero?
3. How does the determinant affect the solutions of a linear system?
4. How can I calculate the determinant of a 3x3 or 4x4 matrix?
5. What is the difference between full rank and deficient rank matrices?

**Tip**: A quadratic equation can be solved using the quadratic formula: \( K = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

For what values of K the matrix [4 4 -3 1; 1 1 -1 0; K 2 2 2; 9 9 K 3] has rank 3

Find the values of K for which the matrix [4 4 -3 1; 1 1 -1 0; K 2 2 2; 9 9 K 3] has rank 3. Explore concepts of linear algebra including matrix rank and determinants. Suitable for undergraduate students.

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