Question

What does a homogenous determinant mean?

Answer 1

Determinants are used to solve a set of homogeneous linear equations,

A homogeneous system of linear equations is one in which all of the constant terms are zero. It is important to note that when we represent a homogeneous system as a matrix, we often leave off the final column of constant terms, since applying row operations would not modify that column.

Ex:

If $2x+3y=4$ and $-x+y=9$ are two homogenous equations.

Matrix notation is,

$\left[\begin{array}{cc}2& 3\\ -1& 1\end{array}\right]$ $\left[\begin{array}{c}x\\ y\end{array}\right]$$=\left[\begin{array}{c}4\\ 9\end{array}\right]$

To solve these homogeneous linear equations we have to use the determinant and which should not be zero.

i.e., $\left|\begin{array}{cc}2& 3\\ -1& 1\end{array}\right|\ne 0$