A determinant is a numerical value that can be calculated from a square matrix. It is used to determine important properties of the matrix, such as whether it is invertible or singular.
The determinant of a matrix can be found by following a specific set of steps, depending on the size of the matrix. For a 2x2 matrix, you can simply multiply the diagonal elements and subtract the product of the off-diagonal elements. For larger matrices, you can use various methods such as cofactor expansion, row reduction, or using the Leibniz formula.
In this context, "det of 4A-1" refers to the determinant of the matrix 4A-1. This means that the given matrix has been multiplied by the scalar value of 4 and then subtracted by the identity matrix (a matrix with all diagonal elements equal to 1 and all other elements equal to 0).
The determinant of a matrix is important because it can provide valuable information about the matrix. For example, a non-zero determinant indicates that the matrix is invertible, which means it has a unique solution for a system of linear equations. It can also be used to calculate the area or volume of a geometric shape represented by 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. For example, a 2x2 matrix with a negative determinant would have a negative product of the diagonal elements and a positive product of the off-diagonal elements.