If the system 2 x-3 y+z=p,-x+2 y+3 z=q, 5 x-8 y-z=r has a solution, then (a) p, q, r are in AP (

If the system 2 x-3 y+z=p,-x+2 y+3 z=q, 5 x-8 y-z=r has a...

Key Concepts

Arithmetic Progression

Arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. Recognizing an arithmetic progression involves verifying that the middle term is the average of the first and last terms. This concept can arise in various contexts, including the relationships among constants in a system of equations when the consistency condition imposes a specific linear relation among them.

Consistency Condition

The consistency condition determines whether a system of linear equations has at least one solution. For systems with a singular coefficient matrix (one with a zero determinant), a necessary condition for consistency is that the constant terms must satisfy the same linear relationship as exists among the corresponding rows of the coefficient matrix. This prevents the augmented system from having a higher rank than the coefficient matrix.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. Understanding such systems is fundamental to linear algebra, as it involves determining whether there exists a set of values for the variables that simultaneously satisfies all the equations. This study includes exploring unique solutions, infinitely many solutions, or the absence of solutions.

Coefficient Matrix and Determinant

For a system of linear equations, the coefficient matrix is formed by the coefficients of the variables. Its determinant is crucial because a nonzero determinant generally indicates that the equations are independent and a unique solution exists. Conversely, a determinant of zero suggests potential dependencies among equations, necessitating further analysis for consistency.

Row Dependency and Rank

When the rows (or equations) of a coefficient matrix are linearly dependent, one row can be expressed as a linear combination of others. This reduces the rank of the matrix below the number of equations. The rank, indicating the number of linearly independent equations, plays a key role in determining the consistency of the system—that is, whether the system has solutions.

Transcript

00:01 So in the given question we are told that if the given system of equations which are given as 2x minus 3y plus z equal to p and minus x minus x plus 2y minus x plus 2y plus 2y plus 3 z is equal to q and and 5x minus 8y minus z is equal to r if this system of equations has a solution has a solution then we are told to find which of the given options is the correct answer right so in the question what we have have is we can take the options as p, q and r are in an ap, r in an arithmetic progression.

01:22 Option b says that pqr are in a gp.

01:31 Option c says pqr, r in an hp, which is a harmonic progression and option d says that q, p are in an ap, right? so which of this option is correct is what we need to find? so a system of equations would, system of equations would have a solution if the determinant, the determinant of the coefficient matrix is not equal to 0 or if we are taking the determinant delta x or we have delta x delta y and delta z and these determinant would be equal to 0 so the system has the has a solution in either if the systems are the equation satisfy either of these cases right so we will discuss what these determinants are when we are solving the problem.

02:42 So first let's find the determinant of the determinant of the coefficient matrix, right? so a is a matrix that is formed with the coefficients of the variables in the equation which would be equal to 2 minus 3 1 minus 1 2 2 3 and 5 minus 8 minus 1.

03:12 So when we take the determinant we have 2 times minus 2 plus 24 minus of minus 3 is plus 3 times 1 minus 15 plus we have 1 times 8 minus 10 and when we evaluate this what we get is my 24 minus 2 is 22 22 times 2 is 44 plus 3.

03:44 3 times minus 14 is minus 42 and 8 minus 10 is minus 2.

03:54 So we would get 0, right? so determinant of a is 0.

03:58 So now we have to find the next condition which is we have to see whether the determinants delta x, delta y or delta z is equal to 0.

04:10 So when we are taking the determinant delta x, delta x, delta x is for...

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