Definition of Determinant

Basic Concepts of Determinant

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:

\det : \mathbb{R}^{n \times n} \to \mathbb{R} : A \mapsto \det A

This function is unique because it has three characteristic properties that no other function has.

Three Determining Properties of Determinant

Linear Property on Each Row

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:

\det \begin{pmatrix} a_1 \\ \vdots \\ a_i + a_i' \\ \vdots \\ a_n \end{pmatrix} = \det \begin{pmatrix} a_1 \\ \vdots \\ a_i \\ \vdots \\ a_n \end{pmatrix}

+ \det \begin{pmatrix} a_1 \\ \vdots \\ a_i' \\ \vdots \\ a_n \end{pmatrix}

When we multiply one row by a scalar:

\det \begin{pmatrix} a_1 \\ \vdots \\ \lambda \cdot a_i \\ \vdots \\ a_n \end{pmatrix}

= \lambda \cdot \det \begin{pmatrix} a_1 \\ \vdots \\ a_i \\ \vdots \\ a_n \end{pmatrix}

Antisymmetric Property

Determinant is antisymmetric, if there are two identical rows, the determinant immediately becomes zero:

\det \begin{pmatrix} \vdots \\ a \\ \vdots \\ a \\ \vdots \end{pmatrix} = 0

This makes sense because if two rows are the same, the matrix cannot have full rank.

Normalization Property

Determinant is normalized so that the determinant of the identity matrix is always 1:

\det I = 1

Where $I$ is an identity matrix of any size.

Matrix Operations and Determinant

If we multiply the entire matrix by a scalar $\lambda$ , the determinant will be affected by the power of n:

\det(\lambda A) = \lambda^n \cdot \det A

This is because each row is multiplied by $\lambda$ , and there are $n$ total rows.

When we swap two rows of a matrix, the determinant changes sign:

\det B = -\det A

Interestingly, when we add a multiple of one row to another row, the determinant doesn't change:

\det \begin{pmatrix} \vdots \\ a_j \\ \vdots \\ a_i + \lambda a_j \\ \vdots \end{pmatrix}

= \det \begin{pmatrix} \vdots \\ a_j \\ \vdots \\ a_i \\ \vdots \end{pmatrix}

Multiplying a row by scalar $\lambda \neq 0$ changes the determinant to $\det A' = \lambda \cdot \det A$ . Swapping rows changes the sign to $\det A' = -\det A$ . Adding multiples of other rows doesn't change the determinant at all.

Relationship with Matrix Invertibility

Determinant is key to understanding whether a matrix can be inverted. For a square matrix A, the following conditions are equivalent:

Matrix A is invertible
There exists an inverse matrix $A^{-1}$ that satisfies $AA^{-1} = I$
Full rank matrix: $\text{rank}(A) = n$
Trivial kernel: $\ker(A) = \{0\}$
Columns are linearly independent
Rows are linearly independent
Determinant is not zero: $\det A \neq 0$

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.

Multiplication Properties and Similar Matrices

One of the most useful properties of determinant is how it interacts with matrix multiplication:

\det(AB) = \det A \cdot \det B

If matrix $A$ is invertible, the determinant of its inverse is:

\det(A^{-1}) = \frac{1}{\det A}

Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $S$ such that:

B = S^{-1}AS

Similar matrices have the same determinant. The proof is simple:

\det B = \det(S^{-1}AS)

= \det(S^{-1}) \cdot \det A \cdot \det S

= \frac{1}{\det S} \cdot \det A \cdot \det S = \det A

When performing Gaussian elimination with $p$ row swaps, the determinant of the resulting matrix $R$ is:

\det R = (-1)^p \cdot \det A

This provides a practical method for calculating determinants.

Common Mistakes About Determinant

It's important to remember that determinant is not additive:

\det(A + B) \neq \det A + \det B

As a simple example:

\det\left(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\right)

= \det\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = 4

But:

\det\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \det\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

= 1 + 1 = 2

Clearly $4 \neq 2$ , so determinant is not additive on matrix addition.

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