if the rank of A is 3 then find p value where A=
[2 2 -3 1] 1ST ROW
[1 1 -1 0] 2ND ROW
[P 2 3 2] 3RD ROW
[4 4 P 3] 4TH ROW

Given that the matrix \( A \) is:

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

and the rank of \( A \) is 3, we need to find the value of \( p \).

### Steps:
1. **Condition for Rank**: For the matrix to have rank 3, it means that one of its \( 3 \times 3 \) submatrices should have a non-zero determinant, while the determinant of the full \( 4 \times 4 \) matrix should be zero.
   
2. **Calculation of Determinant**: We calculate the determinant of the \( 4 \times 4 \) matrix and set it equal to zero to find \( p \).

Let’s calculate the determinant of the matrix \( A \):

\[
\text{det}(A) = \begin{vmatrix} 
2 & 2 & -3 & 1 \\
1 & 1 & -1 & 0 \\
p & 2 & 3 & 2 \\
4 & 4 & p & 3 
\end{vmatrix}
\]

Now, I will compute the determinant and solve for \( p \).The determinant of matrix \( A \) is given by \( p^2 + 5p - 14 \). Setting this determinant to zero, we get:

\[
p^2 + 5p - 14 = 0
\]

Solving this quadratic equation, the possible values of \( p \) are:

\[
p = -7 \quad \text{or} \quad p = 2
\]

Thus, the values of \( p \) that satisfy the rank condition are \( p = -7 \) or \( p = 2 \).

Would you like more details on how this was solved, or do you have any other questions?

Here are some related questions:
1. How do we compute the rank of a matrix from its row echelon form?
2. How do different values of \( p \) affect the rank of the matrix?
3. What are some methods to calculate the determinant of a larger matrix?
4. Can you explain the significance of the determinant being zero for the matrix rank?
5. How do we determine the null space of a matrix?

**Tip**: To compute the rank of a matrix, you can also reduce it to row echelon form and count the non-zero rows.

If the rank of A is 3 then find p value where A = [2 2 -3 1] 1ST ROW, [1 1 -1 0] 2ND ROW, [P 2 3 2] 3RD ROW, [4 4 P 3] 4TH ROW.

Learn how to determine the value of p for a 4x4 matrix with a rank of 3. This problem involves matrix rank, determinants, and solving quadratic equations, making it suitable for undergraduate students studying linear algebra.

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