Math Problem Statement
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
Solution
To determine the values of such that the rank of matrix is 3, we first need to compute the determinant of the submatrices of and ensure that the matrix has exactly 3 linearly independent rows or columns.
The matrix is:
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$$. --- 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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Determinants
Formulas
Matrix determinant formula
Theorems
Properties of determinants
Suitable Grade Level
Advanced College