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 the matrix is 3, we need to analyze the matrix and ensure that it has three linearly independent rows (or columns). The matrix is given by:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Theory
Linear Algebra
Determinants
Rank of a Matrix
Quadratic Equations
Formulas
Determinant of a 4x4 matrix
Theorems
Rank-Nullity Theorem
Suitable Grade Level
College