Question

every homogeneous system of linear equations is...​

Answer 1

Answer:

Every homogeneous system has at least one solution, known as the zero (or trivial) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix (det(A) ≠ 0) then it is also the only solution.

Answer 2

Answer:

Every homogeneous system of linear equation is always consistent.

Explanation:

• In homogeneous system of linear equation in which the linear equations have no constant term. Linear equation means it contains one or more equation which has same variables.
• It is said to be Homogeneous system of linear equation if all the constant terms are Zero.

Example:

$a x_{1} + by_{1} + c z_{1} = 0\\ a x_{2} + b y_{2} + c z_{2} = 0\\ a x_{3} + b y_{3} + c z_{3} = 0$

• By using the determinant of coefficient matrix we can found out whether the homogeneous linear solution has unique solution or infinite number of solutions.
• Unique solution is known as trivial solution and infinite number of solution is known as non trivial solution.

Properties of homogeneous system of linear equation:

• The homogeneous system contains at least one solution and it is known as trivial solution.
• The sum of the solution is given as a+b where a and b are two solution of homogeneous linear system.
• It has infinitely many solution if at least one free is found on the homogeneous system.
• If the homogeneous system has more unknowns than the equations, therefore the system has infinitely many solutions.
• The zero vector will always the solution of the homogeneous system.
• Evey homogeneous system of linear equation is always consistent.

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