In Exercises 57–60, solve each equation for x. |1 x - 2||3 1 1| = - 8|0 - 2 2|

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation it represents. For a 3x3 matrix, the determinant can be calculated using a specific formula involving the elements of the matrix.

Matrix equations involve expressions where matrices are set equal to each other or to a scalar. In this case, the equation represents a determinant of a matrix set equal to -8. Solving such equations often requires manipulating the matrix to isolate the variable, which in this case is 'x'.

Solving for variables in equations involves isolating the variable of interest through algebraic manipulation. In the context of matrix equations, this may include expanding the determinant, simplifying the resulting expression, and then solving for the unknown variable, ensuring that all steps adhere to algebraic principles.