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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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9. 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.
Transcribed Image Text:9. 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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