### Matrix calculus: Determinants

### 2-dimensional determinants

Previously we saw that an #(n\times n)#-matrix #A# of rank #n# has some nice properties: according to theorem *Invertibilty and rank* the inverse of #A# exists and according to *Unique solution and maximal rank* each system of linear equations with coefficient matrix #A# has exactly one solution.

We also discussed that determining the rank of a matrix is possible by reducing rows and columns to the reduced echelon form and counting the number of independent rows (or columns) of the result. We will however also be able to see whether the rank is #n# or not by calculating the so-called *determinant* of the #(n\times n)#-matrix*.* The calculation of a this number is fairly simple, but the underlying theory is more complicated.

Determinant of a two-dimensional square matrix

The determinant of a #(2\times 2)#-matrix

\[

A=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)

\] is the number #a\cdot d-b\cdot c#.

The usual notation for the determinant of #A# is #\det(A)# or \(

\left|\begin{array}{cc} a & b \\ c & d \end{array}\right|\).

Determinant criterion for invertibility The linear map #\mathbb{R}^2\to\mathbb{R}^2# with matrix #A# is invertible if and only if #\det(A)\ne0#.

In this case, the inverse of #A# is equal to

\[A^{-1} =\dfrac{1}{\det(A)}\cdot \matrix{d&-b\\ -c&a}\]

The expression #a\cdot d-b\cdot c# is not linear, and thus appears to fall outside the scope of linear algebra. Fortunately, the expression is multilinear. Certain properties of the #(2\times 2)#-determinant model generalize to #(n\times n)#-determinants.

Characteristic properties of the determinant Regard the determinant as a function of the two rows of the matrix: #\det(\vec{a}_1, \vec{a}_2)#. Then #\det# satisfies the following three properties (for all choices of vectors and scalars):

**bilinearity**(linearity in both arguments):

\[

\begin{array}{rcl}

\det(\lambda_1 \vec{b}_1 +\lambda_2 \vec{b}_2, \vec{a}_2)&=&

\lambda_1 \det(\vec{b}_1, \vec{a}_2) + \lambda_2 \det(\vec{b}_2 ,\vec{a}_2)\\

\det(\vec{a}_1, \lambda_1 \vec{b}_1 +\lambda_2 \vec{b}_2)&=&

\lambda_1 \det(\vec{a}_1, \vec{b}_1) + \lambda_2\det(\vec{a}_1 ,\vec{b}_2)

\end{array}

\]**antisymmetry**: #\det(\vec{a}_1,\vec{a}_2)=-\det(\vec{a}_2,\vec{a}_1)# (swapping two vectors produces a minus sign); as a consequence, if the two vectors are identical to each other, then #\det(\vec{a},\vec{a})=0#;**normalization**: #\det(\vec{e}_1,\vec{e}_2)=1#.

The determinant is unique in the sense that, if a function of pairs of vectors has the above three properties of #\det#, it is the determinant function #\det#.

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