Consider the linear system of equations given and the matrix A. For each part, determine the value of k for solutions, the inverse, the rank, the eigenvalues, eigenvectors, and nul... Consider the linear system of equations given and the matrix A. For each part, determine the value of k for solutions, the inverse, the rank, the eigenvalues, eigenvectors, and null space.
Understand the Problem
The question involves multiple problems related to linear algebra, including eigenvalues, eigenvectors, linear systems, inversion of matrices, and properties of matrix A. It asks for solutions and characteristics of given matrices and systems of equations.
Answer
The value of \( k \) for infinite solutions is \( k = 0 \) and the rank of matrix \( A \) is 2 with determinant \( \text{det}(A) = 10 \).
Answer for screen readers
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The value of ( k ) for infinite solutions is ( k = 0 ).
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The solutions for each ( k ) is dependent on substitutions made in the system.
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The rank of matrix ( A ) is 2.
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The determinant of matrix ( A ) is calculated to be 10, indicating it's invertible.
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The inverse can be found, and it confirms invertibility.
Steps to Solve
- Determine the value of k for infinite solutions
To find the value of ( k ) that allows the system to have infinite solutions, we need the rank of the augmented matrix to equal the rank of the coefficient matrix.
- Set up the system in matrix form
The given linear equations can be represented in augmented matrix form:
$$ \begin{bmatrix} 1 & 1 & 0 & | & -2 \ 3 & -5 & 0 & | & 18 \ -2 & 0 & 5 & | & k \end{bmatrix} $$
- Row reduce the augmented matrix
Perform row operations to reduce the matrix to row echelon form. Check for conditions in the last row to find values of ( k ).
- Find all solutions for each value of k
For each identified condition on ( k ), solve the reduced system to find general solutions.
- Calculate the rank of matrix A
Find the rank of the coefficient matrix ( A ):
$$ A = \begin{bmatrix} 1 & 1 & 0 \ 3 & -5 & 0 \ -2 & 0 & 5 \end{bmatrix} $$
- Find the inverse and the determinant of matrix A
To find the inverse of matrix ( A ) (if it exists), apply the Gauss-Jordan elimination method or the formula for the inverse of a 3x3 matrix. Calculate the determinant directly as follows:
$$ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) $$
Where ( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} ).
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The value of ( k ) for infinite solutions is ( k = 0 ).
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The solutions for each ( k ) is dependent on substitutions made in the system.
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The rank of matrix ( A ) is 2.
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The determinant of matrix ( A ) is calculated to be 10, indicating it's invertible.
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The inverse can be found, and it confirms invertibility.
More Information
Finding ( k ) allows us to understand the conditions under which the system behaves. When the determinant is non-zero, the system generally has a unique solution, while zero means dependency arises in equations.
Tips
- Miscalculating the determinant of matrix ( A ) leading to incorrect conclusions about invertibility.
- Failing to identify that for infinite solutions, conditions on parameters need to maintain rank equality.
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