Infinitely Many Solutions? A Homework Problem to Investigate

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SUMMARY

The discussion centers on the mathematical problem regarding singular matrices and their solutions. It establishes that if A is a singular n by n matrix, the equation A*x=b can have infinitely many solutions depending on the determinant of the matrix and the vector b. Specifically, if b=0, then A*x=0 has infinitely many solutions when det(A)=0. Conversely, if b is non-zero, the number of solutions can vary based on the determinant of the augmented matrix formed by replacing the nth column of A with b.

PREREQUISITES
  • Understanding of singular matrices and their properties
  • Knowledge of determinants and their implications in linear algebra
  • Familiarity with linear equations and solution sets
  • Basic concepts of augmented matrices in solving systems of equations
NEXT STEPS
  • Study the properties of singular matrices in linear algebra
  • Learn about determinants and their role in determining the number of solutions
  • Explore the concept of augmented matrices and their applications
  • Investigate the implications of different cases for vector b in linear equations
USEFUL FOR

Students studying linear algebra, mathematics educators, and anyone interested in solving systems of linear equations involving singular matrices.

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I had a homework problem and i was wondering if anyone could help me out with it. It is a true or fasle question that requires explanation whether true or false. It goes like this...If A is a singular n by n matrix, then A*x=b has infinitely many solutions.(True or False)
 
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Is the matrix N = (0) singular? Does Nx = b have infinitely many solutions if b is nonzero?
 
There are several case :

b=0, then Ax=0 has infinitely many solution if det(A)=0

b<>0 then it can have zero or an infinity of solution, depending on the determinants : det(b|_nA)

where b|_nA means : the nth column of A is replace by b.

So the answer is "true and false" because you don't specify enough the question.
 
or you might say the question is meaningless since you do not quantify the letter "b".
 

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