A matrix is a rectangular array of numbers or symbols that are arranged in rows and columns. It is used in mathematics to represent and solve systems of equations, transformations, and other mathematical operations.
A determinant is a numerical value that is calculated from the elements of a square matrix. It is significant in matrix operations because it can determine whether a matrix has a unique solution, and it is also used to find the inverse of a matrix.
The determinant of a 2x2 matrix is calculated by multiplying the elements in the main diagonal and subtracting the product of the elements in the other diagonal. For example, the determinant of the matrix [a b; c d] is ad - bc.
The absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the column vectors of the matrix. Similarly, the absolute value of the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by the column vectors of the matrix.
Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the arrangement of the elements in the matrix and does not affect its numerical value.