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Determinant is a very important mathematical tool in linear algebra. Think of determinant as a "strength" measure of a matrix, it tells us how much change a space undergoes when transformed by that matrix.
Every square matrix has one unique determinant value. This value can be positive, negative, or zero, and each tells us different information about the matrix.
Determinant is a special function that takes a square matrix and produces one real number:
This function is unique because it has three characteristic properties that no other function has.
Determinant is linear on each row of the matrix. This means, if we change one row, the determinant will change linearly.
When we add two vectors in one row:
When we multiply one row by a scalar:
Determinant is antisymmetric, if there are two identical rows, the determinant immediately becomes zero:
This makes sense because if two rows are the same, the matrix cannot have full rank.
Determinant is normalized so that the determinant of the identity matrix is always :
Where is an identity matrix of any size.
If we multiply the entire matrix by a scalar , the determinant will be affected by the power of n:
This is because each row is multiplied by , and there are total rows.
When we swap two rows of a matrix, the determinant changes sign:
Interestingly, when we add a multiple of one row to another row, the determinant doesn't change:
Multiplying a row by scalar changes the determinant to . Swapping rows changes the sign to . Adding multiples of other rows doesn't change the determinant at all.
Determinant is key to understanding whether a matrix can be inverted. For a square matrix A, the following conditions are equivalent:
If the determinant is zero, the matrix "flattens" space to a lower dimension, so the transformation cannot be inverted.
If the rows of a matrix are linearly dependent, the determinant must be zero. This happens because of the antisymmetric property of determinant, linear dependence creates a situation where we can make identical rows through linear operations.
One of the most useful properties of determinant is how it interacts with matrix multiplication:
If matrix is invertible, the determinant of its inverse is:
Two matrices and are called similar if there exists an invertible matrix such that:
Similar matrices have the same determinant. The proof is simple:
When performing Gaussian elimination with row swaps, the determinant of the resulting matrix is:
This provides a practical method for calculating determinants.
It's important to remember that determinant is not additive:
As a simple example:
But:
Clearly , so determinant is not additive on matrix addition.
