In geometry, what term is used for the value obtained by manipulating a matrix?

The term used for the value obtained by manipulating a matrix is the determinant. The determinant is a scalar value that can be computed from the elements of a square matrix and provides important insights about the properties of the matrix. For example, the determinant can indicate whether a matrix is invertible; if the determinant is zero, the matrix does not have an inverse. Additionally, the determinant is used in various applications such as solving systems of linear equations, and it provides information about the volume scaling factor of linear transformations represented by the matrix.

In contrast, coefficients refer to the numerical factors that multiply variables in algebraic expressions, factors are numbers or expressions that can be multiplied together to get another number or expression, and variables are symbols used to represent numbers in mathematical expressions. These terms do not pertain specifically to the value computed from a matrix and therefore are not the correct choice in this context.