
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 matrix \( A \) is 3, we first need to compute the determinant of the \( 3 \times 3 \) submatrices of \( A \) and ensure that the matrix \( A \) has exactly 3 linearly independent rows or columns.

The matrix \( A \) is:

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

We will select \( 3 \times 3 \) submatrices from \( A \) and calculate their determinants. The rank will be 3 if at least one of these determinants is non-zero for a particular value of \( p \).

Let's consider the following \( 3 \times 3 \) submatrix formed by the first three rows and columns of \( A \):

\[
M_1 = \begin{pmatrix}
1 & 1 & -1 \\
4 & 4 & -3 \\
p & 2 & 2
\end{pmatrix}
\]

The determinant of \( M_1 \) is:

\[
\text{det}(M_1) = 1 \cdot \begin{vmatrix} 4 & -3 \\ 2 & 2 \end{vmatrix} - 1 \cdot \begin{vmatrix} 4 & -3 \\ p & 2 \end{vmatrix} - 1 \cdot \begin{vmatrix} 4 & 4 \\ p & 2 \end{vmatrix}
\]

Let's calculate each of the smaller determinants:

\[
\begin{vmatrix} 4 & -3 \\ 2 & 2 \end{vmatrix} = 4 \cdot 2 - (-3) \cdot 2 = 8 + 6 = 14
\]

\[
\begin{vmatrix} 4 & -3 \\ p & 2 \end{vmatrix} = 4 \cdot 2 - (-3) \cdot p = 8 + 3p = 8 + 3p
\]

\[
\begin{vmatrix} 4 & 4 \\ p & 2 \end{vmatrix} = 4 \cdot 2 - 4 \cdot p = 8 - 4p
\]

Substituting these into the determinant of \( M_1 \):

\[
\text{det}(M_1) = 1 \cdot 14 - 1 \cdot (8 + 3p) - 1 \cdot (8 - 4p)
\]

Simplifying:

\[
\text{det}(M_1) = 14 - 8 - 3p - 8 + 4p = -2 + p
\]

So:

\[
\text{det}(M_1) = p - 2
\]

The determinant \( \text{det}(M_1) \) will be non-zero when \( p \neq 2 \).

Therefore, \( p \neq 2 \) will ensure that the rank of \( A \) is 3.

### Final Answer:
The values of \( p \) such that the rank of \( A \) is 3 are all values \( p \) except \( p = 2 \).

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Would you like further details on this process or do you have any questions?

Here are some related questions:
1. What does the rank of a matrix represent in linear algebra?
2. How can the determinant of a matrix help determine the rank?
3. What is the significance of linearly independent rows in a matrix?
4. Can the rank of a matrix be greater than its number of rows or columns?
5. What happens to the rank if a matrix is multiplied by a non-zero scalar?

**Tip:** Remember that the rank of a matrix is the maximum number of linearly independent rows or columns, which determines the dimensionality of the space spanned by the matrix.

Explore how to determine values of p such that the rank of matrix A is 3 with a detailed step-by-step solution. Understand the importance of matrix rank, determinants, and linear independence in linear algebra. Suitable for advanced college students and professionals.

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