Three planes can fail to have an intersection point, when no...

Three planes can fail to have an intersection point, when no two planes are parallel The system is singular if row 3 of $A$ is a __________ of the first two rows. Find a third equation that can't be solved if $x+y+z=0$ and $x-2 y-z=1$

Key Concepts

Singular Matrix

A matrix is singular if it does not have full rank, meaning its rows (or columns) are linearly dependent. This results in a determinant of zero and indicates that the system of equations represented by the matrix may not have a unique solution, and in some cases, can be inconsistent.

Linear Dependence

This concept refers to a situation in a set of vectors (or rows of a matrix) where one vector can be expressed as a combination of others. In the context of systems of linear equations, if one equation (or row) is a linear combination of the others, it does not add new information, and the system becomes singular or has an infinite number of solutions along a particular dimension.

Inconsistency in Systems

An inconsistent system refers to a set of equations that contradict each other, meaning no single solution exists that satisfies all of them simultaneously. In geometric terms, for planes in space, this implies that while pairs of planes might intersect, the third plane does not pass through the line or point of intersection, leading to no common intersection point.

Systems of Linear Equations

This is a collection of linear equations involving the same set of variables. The analysis of these systems involves determining whether a unique solution exists, infinitely many solutions are present, or the system is inconsistent. The concepts of linear dependence and matrix singularity are central in assessing the solvability of these systems.

Geometric Interpretation of Planes

In three-dimensional space, each equation represents a plane. The intersection of two planes typically forms a line; however, adding a third plane leads to different scenarios. If the third plane is positioned such that it does not intersect the line of intersection of the first two (or is a linear combination of the first two), the system can result in no common intersection, thereby having no solution.

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