For the equation system x+y+z=b², x + ay + z = b, x + a²y-z=b³, write down the matrix of coefficients and the augmented matrix. Find the determinant of the matrix of coefficients. Find also the rank of the matrix of coefficients and the rank of the augmented matrix for all real numbers a, b. The set S comprises all the triples (x, y, z) which satisfy the above equation system. Determine for which pairs of values of a and b (if any) the set S (i) contains exactly one element, (ii) is empty, (iii) is finite but contains more than one element, and (iv) is infinite. In the case where there is a unique solution, find it.
For the equation system x+y+z=b², x + ay + z = b, x + a²y-z=b³, write down the matrix of coefficients and the augmented matrix. Find the determinant of the matrix of coefficients. Find also the rank of the matrix of coefficients and the rank of the augmented matrix for all real numbers a, b. The set S comprises all the triples (x, y, z) which satisfy the above equation system. Determine for which pairs of values of a and b (if any) the set S (i) contains exactly one element, (ii) is empty, (iii) is finite but contains more than one element, and (iv) is infinite. In the case where there is a unique solution, find it.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.1: The Determinants Of A Matrix
Problem 70E: The determinant of a 22 matrix involves two products. The determinant of a 33 matrix involves six...
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For the equation system
x+y+z=b²,
x + ay + z = b,
x + a²y-z = b³,
write down the matrix of coefficients and the augmented matrix. Find the determinant of the matrix
of coefficients. Find also the rank of the matrix of coefficients and the rank of the augmented matrix
for all real numbers a, b.
The set S comprises all the triples (x, y, z) which satisfy the above equation system. Determine for
which pairs of values of a and b (if any) the set S (i) contains exactly one element, (ii) is empty,
(iii) is finite but contains more than one element, and (iv) is infinite.
In the case where there is a unique solution, find it.
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