Statement 1 The system of equations x+2 y+3 z=1, x-y+4 z=0,2...

Statement 1 The system of equations $x+2 y+3 z=1, x-y+4 z=0,2 x+y+7 z=1$ has infinitely many solutions. and Statement 2 System of equations $\mathrm{AX}=\mathrm{B}$ is inconsistent if $|\mathrm{A}|=0$ and $(\operatorname{adj}$ A) $\mathrm{B} \neq 0$ Suppose $\mathrm{x}$ is the weight in $\mathrm{kg}, \mathrm{y}$ is the height in $\mathrm{cm}$ and $\mathrm{z}$ is the waist measurement in $\mathrm{cm} .$ A dietician wants to study the relationship between $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ assuming that there is a linear relationship between them. From the data collected on 3 days, he obtained the following equations, satisfied by $x, y$, and $z .$ $$ \begin{array}{l} 2 x+p y+6 z=8 \\ x+2 y+q z=5 \\ x+y+3 z=4 \end{array} $$ where, $\mathrm{p}$ and $\mathrm{q}$ are some constants.

Statement 1
The system of equations $x+2 y+3 z=1, x-y+4 z=0,2 x+y+7 z=1$ has infinitely many solutions. and
Statement 2
System of equations $\mathrm{AX}=\mathrm{B}$ is inconsistent if $|\mathrm{A}|=0$ and $(\operatorname{adj}$ A) $\mathrm{B} \neq 0$
Suppose $\mathrm{x}$ is the weight in $\mathrm{kg}, \mathrm{y}$ is the height in $\mathrm{cm}$ and $\mathrm{z}$ is the waist measurement in $\mathrm{cm} .$ A dietician wants to study the relationship between $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ assuming that there is a linear relationship between them. From the data collected on 3 days, he obtained the following equations, satisfied by $x, y$, and $z .$
$$
\begin{array}{l}
2 x+p y+6 z=8 \\
x+2 y+q z=5 \\
x+y+3 z=4
\end{array}
$$
where, $\mathrm{p}$ and $\mathrm{q}$ are some constants.

Key Concepts

Linear Systems of Equations

A collection of equations involving the same set of variables is known as a linear system. For such systems, the goal is to find values for the variables that satisfy all the equations simultaneously. The behavior of solutions—whether a unique solution, infinitely many solutions, or no solution exists—depends on the relationships between the equations, such as their coefficients and constants.

Consistency and Inconsistency of Systems

The concepts of consistency and inconsistency determine whether a system of linear equations has at least one solution or none at all. A system is consistent if there is at least one set of values for the variables that satisfies every equation, and it is inconsistent if no such set exists. Conditions on determinants and matrix ranks often play a crucial role in these determinations.

Matrix Determinant and Invertibility

The determinant of a coefficient matrix in a system of linear equations is a key indicator of the system’s behavior. If the determinant is non-zero, the matrix is invertible, and the system typically has a unique solution. Conversely, if the determinant is zero, the matrix is singular, and the system may have either infinitely many solutions or be inconsistent depending on additional conditions.

Adjugate Matrix and Singular Systems

The adjugate (or classical adjoint) of a matrix is used in determining the inverse of a matrix through the formula involving the determinant. In cases where the determinant of the matrix is zero, the adjugate can be employed to check for consistency. Specifically, if the product of the adjugate matrix with the right-hand side vector does not vanish, it can indicate that the system is inconsistent.

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