The Value of Determinant

### Program Information

Name: The Value of Determinant
Domain: Numerical program
Functionality: Compute determinant of the matrices
Input: $\mathbb{R}_{n,m}$ is the set of all matrices with real elements, $n$ rows, and $m$ columns. A matrix $A\in \mathbb{R}_{n,m}$ with $$\mathbb{A}=\left(\begin{array}{ccc} a_{1,1} & \ldots & a_{1,m} \\ \vdots & \ddots & \vdots \\ a_{n,1} & \ldots & a_{n,m} \end{array} \right)$$ can also be expressed by its column vectors $A=(a^1,\cdots ,a^m)$ resp. its row vectors $A=(a_1,\cdots ,a_n)^T$,where $A^T$ denotes the $transposition$ of matrix $A$
Output: For a matrix $A=(a_{i,j})_{1 \leq i,j \leq n} \in \mathbb{R}_{n,n}$ let $det(A)=|A|$ denote the $determinant$ of $A$ which is defined as the scalar  $$det(A)=\left|\begin{array}{ccc} a_{1,1} & \ldots & a_{1,n} \\ \vdots & \ddots & \vdots \\ a_{n,1} & \ldots & a_{n,n} \end{array} \right| := \sum_{\pi \in S_n}sgn(\pi) \prod_{j=1}^{n}a_{j,\pi(j)}$$ where $S_n$ is the permutation group on $\{1,\cdots ,n\}$, and $sgn(\pi)$ denotes the $sign$ of permutation $\pi$ The determinant $det(A_{i,j})$ of a submatrix $A_{i,j}\in \mathbb{R}_{n_1,n_1}$ of $A \in \mathbb{R}_{n,m}$ is called a $minor$ of $A$.

#### Reference

An Empirical Study on the Selection of Good Metamorphic Relations https://doi.org/10.1109/COMPSAC.2006.24

### MR Information

#### MR1------  Transposition

Description:
Property: $det(A)=det(A^T)$
Source input: $A$
Source output: $det(A)$
Follow-up input: $A^T$
Follow-up output: $det(A^T)$
Input relation: $A \Rightarrow A^T$
Output relation: $det(A)=det(A^T)$
Pattern:

#### MR2------ Exchange of Rows

Description:
Property: $-det((a_1,\dots ,a_n)^T)=det((a_1,\dots ,a_{j-1},a_i,a_{j+1},\dots ,a_{i-1},a_j,a_{i+1},\dots ,a_n)^T)$ for $i,j \in \{1,\dots ,n\},j < i$
Source input: $A=(a_1,\dots ,a_n)^T$
Source output: $det(A)$
Follow-up input: $B=(a_1,\dots ,a_{j-1},a_i,a_{j+1},\dots ,a_{i-1},a_j,a_{i+1},\dots ,a_n)^T$
Follow-up output: $det(B)$
Input relation: $A \Rightarrow B$ for $i,j \in \{1,\dots ,n\},j < i$
Output relation: $-det(A)=det(B)$
Pattern:

#### MR3------ Exchange of Columns

Description:
Property: $-det((a^1,\dots ,a^n))=det((a^1,\dots ,a^{j-1},a^i,a^{j+1},\dots ,a^{i-1},a^j,a^{i+1},\dots ,a^n))$ for $i,j \in \{1,\dots,n\},j < i$
Source input:  $A=(a^1,\dots ,a^n)$
Source output: $det(A)$
Follow-up input:  $B=(a^1,\dots ,a^{j-1},a^i,a^{j+1},\dots ,a^{i-1},a^j,a^{i+1},\dots ,a^n)$
Follow-up output: $det(B)$
Input relation: $A \Rightarrow B$ for $i,j \in \{1,\dots,n\},j < i$
Output relation: $-det(A)=det(B)$
Pattern:

#### MR4------ Row Multiplied with Scalar

Description:
Property: $\beta det((a_1,\dots ,a_{k-1},a_k,a_{k+1},\dots ,a_n)^T)=det((a_1,\dots ,a_{k-1},\beta a_k,a_{k+1},\dots ,a_n)^T)$ for $k \in \{1,\dots ,n\}$
Source input:  $A=(a_1,\dots ,a_{k-1},a_k,a_{k+1},\dots ,a_n)^T$
Source output: $det(A)$
Follow-up input: $B=(a_1,\dots ,a_{k-1},\beta a_k,a_{k+1},\dots ,a_n)^T$
Follow-up output:  $det(B)$
Input relation: $A \Rightarrow B$ for $k \in \{1,\dots ,n\}$
Output relation: $\beta det(A)=det(B)$
Pattern:

#### MR5------ Column Multiplied with Scalar

Description:
Property: $\beta det((a^1,\dots ,a^{k-1},a^k,a^{k+1},\dots ,a^n)^T)=det((a^1,\dots ,a^{k-1},\beta a^k,a^{k+1},\dots ,a^n)^T)$ for $k \in \{1,\dots ,n\}$
Source input: $A=(a^1,\dots ,a^{k-1},a^k,a^{k+1},\dots ,a^n)^T$
Source output:  $det(A)$
Follow-up input:  $B=(a^1,\dots ,a^{k-1},\beta a^k,a^{k+1},\dots ,a^n)^T$
Follow-up output:  $det(B)$
Input relation: $A \Rightarrow B$ for $k \in \{1,\dots ,n\}$
Output relation: $\beta det(A)=det(B)$
Pattern:

#### MR6------ Addition of Rows of Two Matrices

Description:
Property: $det((a_1,\dots ,a_{k-1},a_k + b_k,a_{k+1},\dots ,a_n)^T)=det((a_1,\dots ,a_{k-1}, a_k,a_{k+1},\dots ,a_n)^T) + det((a_1,\dots ,a_{k-1},b_k,a_{k+1},\dots , a_n)^T)$ for $k \in \{1,\dots ,n\}$
Source input: $A=(a_1,\dots ,a_{k-1},a_k + b_k,a_{k+1},\dots ,a_n)^T$
Source output:  $det(A)$
Follow-up input:  $B=(a_1,\dots ,a_{k-1}, a_k,a_{k+1},\dots ,a_n)^T \\C=(a_1,\dots ,a_{k-1},k_k,a_{k+1},\dots , a_n)^T$
Follow-up output:  $det(B),det(C)$
Input relation: $A \Rightarrow B,C$ for $k \in \{1,\dots ,n\}$
Output relation: $det(A)=det(B)+det(C)$
Pattern:

#### MR7------ Addition of Columns of Two Matrices

Description:
Property: $det((a^1,\dots ,a^{k-1},a^k + b^k,a^{k+1},\dots ,a^n)^T)=det((a^1,\dots ,a^{k-1}, a^k,a^{k+1},\dots ,a^n)^T) + det((a^1,\dots ,a^{k-1},b^k,a^{k+1},\dots , a^n)^T)$ for $k \in \{1,\dots ,n\}$
Source input:  $A=(a^1,\dots ,a^{k-1},a^k + b^k,a^{k+1},\dots ,a^n)^T$
Source output:  $det(A)$
Follow-up input:  $B=(a^1,\dots ,a^{k-1}, a^k,a^{k+1},\dots ,a^n)^T \\ C=(a^1,\dots ,a^{k-1},b^k,a^{k+1},\dots , a^n)^T$
Follow-up output:  $det(B),det(C)$
Input relation: $A \Rightarrow B,C$ for $k \in \{1,\dots ,n\}$
Output relation: $det(A)=det(B)+det(C)$
Pattern:

Description:
Property: $det((a_1,\dots ,a_n)^T)=det((a_1,\dots ,a_{j-1},a_j+\beta a_i,a_{j+1},\dots ,a_n)^T)$ for $\beta \in \mathbb{R},i,j\in \{1,\dots ,n\},i \neq j$
Source input: $A=(a_1,\dots ,a_n)^T$
Source output: $det(A)$
Follow-up input:  $B=(a_1,\dots ,a_{j-1},a_j+\beta a_i,a_{j+1},\dots ,a_n)^T$
Follow-up output: $det(B)$
Input relation: $A \Rightarrow B$ for $\beta \in \mathbb{R},i,j\in \{1,\dots ,n\},i \neq j$
Output relation: $det(A)=det(B)$
Pattern:

Description:
Property: $det((a^1,\dots ,a^n)^T)=det((a^1,\dots ,a^{j-1},a^j+\beta a^i,a^{j+1},\dots ,a^n)^T)$ for $\beta \in \mathbb{R},i,j\in \{1,\dots ,n\},i \neq j$
Source input: $A=(a^1,\dots ,a^n)^T$
Source output:  $det(A)$
Follow-up input: $B=(a^1,\dots ,a^{j-1},a^j+\beta a^i,a^{j+1},\dots ,a^n)^T$
Follow-up output: $det(B)$
Input relation: $A \Rightarrow B$ for $\beta \in \mathbb{R},i,j\in \{1,\dots ,n\},i \neq j$
Output relation: $det(A)=det(B)$
Pattern:

#### MR10------ Cofactor Expansion of a Row

Description:
Property: $det(A)=\sum_{j=1}^n (-1)^{i+j}a_{i,j}det(A_{i,j})$ for $i \in \{1,\dots , n\}$
Source input: $A$
Source output: $det(A)$
Follow-up input:  $A_{i,j}$
Follow-up output: $det(A_{i,j})$
Input relation: $A \Rightarrow A_{i,j}$ for $i \in \{1,\dots , n\}$
Output relation: $det(A)=\sum_{j=1}^n (-1)^{i+j}a_{i,j}det(A_{i,j})$
Pattern:

#### MR11------ Cofactor Expansion of a Column

Description:
Property: $det(A)=\sum_{i=1}^n (-1)^{i+j}a_{i,j}det(A_{i,j})$ for $j \in \{1,\dots , n\}$
Source input:  $A$
Source output: $det(A)$
Follow-up input:  $A_{i,j}$
Follow-up output:  $det(A_{i,j})$
Input relation: $A \Rightarrow A_{i,j}$ for $j \in \{1,\dots , n\}$
Output relation: $det(A)=\sum_{j=1}^n (-1)^{i+j}a_{i,j}det(A_{i,j})$
Pattern:

#### MR12------ Multiplication

Description:
Property: $det(A)\cdot det(B)=det(AB)$
Source input:  $A,B$
Source output:  $det(A),det(B)$
Follow-up input: $AB$
Follow-up output: $det(AB)$
Input relation: $A,B \Rightarrow AB$
Output relation: $det(A)\cdot det(B)=det(AB)$
Pattern:
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