Properties of determinants

Determinant of a matrix $A$ is equal to the determinant of its transpose matrix

• Transpose Matrix: $detA = det A^T$
• Triangular Matrix: Equal to the product of the diagonal elements
• Row(Column) swap: If two rows (columns) are swapped the determinant of the matrix $A_s$ is $-detA$
• Null determinant: if two rows(columns) are identical(sum) then $detA=0$
  • If and only if the rows(columns) are linearly dependent
  • if and only if at least one row(column) is a linear combination of the other rows(columns)
  • if a row(column) is proportional to another row(column)
• Invariant Determinant: Row(column) the elements of another row(column) all multiplied by the same scalar are added, the determinant remains the same
• Row(column) multiplied by a scalar: Row(column) is multiplied by $\lambda$ then $\lambda detA$
• Matrix multiplied by a scalar: If $\lambda$ is scalar, $det(\lambda A) = \lambda ^n detA$
• Determinant of the product: Product between two matrices is equal to the products of the determinants. $det(AB) = det(A)det(B) = det(B)det(A) = det(BA)$

Adjugate Matrices

Submatrix

Obtained from $A$ by cancelling $m-r$ rows and $n-s$ columns. Where r,s 2 positive integers such that $1\le r \le m$ and $1\le s\le n$

• Rows dont have to be continuous
• $A=\begin{pmatrix} 3&3&1&0\\ 2&4&1&2\\ 5&1&1&1\\ \end{pmatrix}$
• Can be obtained by cancelling the second row, second and fourth column
• $\begin{pmatrix} 3&1\\ 5&1 \\ \end{pmatrix}$

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• Minor: Determinant of submatrix
• Major: If submatrix is the largest square
• Complement submatrices: Obtained by cancelling on the $i^{th}$ row and the $j^{th}$ column from $A$ to the element $a_{i,j}$
• Complement Minor: Determinant, indicated $M_{i,j}$
• Cofactors: $A_{i,j} = (-1)^{i+j}M_{i,j}$

Adjugate Matrix

$A=\begin{pmatrix} a_{1,1}&a_1,_2&...&a_1,_n \\ a_2,_1&a_2,_2&...&a_2,_n \\ ...&...&...&...\\ a_n,_1&a_n,_2&...&a_n,_n \\ \end{pmatrix}$

Let $A_{i,j}$ be the cofactor for $a_{i,j}$. The adjugate matrix (adjunct or adjoin) $A$ is: $adj(A)=\begin{pmatrix} A_1,_1&A_2,_1&...&A_n,_1 \\ A_1,_2&a_2,_2&...&A_n,_2 \\ ...&...&...&...\\ A_{1,n}&A_{2,n}&...&A_{n,n}\\ \end{pmatrix}$

Dan Terms: Transpose it (Flip i,j around), then calculate the determinate of the remaining rows once removed it(the sub matrix, as would do normally). Then alternate between 1 and -1.(Use cofactors formula!. $(-1)^{i+j}$)

Tutorial Def:

1. $A$
2. $A^T$
3. All Compliment miners
4. Multiple by coefficent = adjudigate matrix
5. $A^{-1} = \frac{1}{detA}\times adj(A)$

Laplace Theorems

Theorem 1

Determinant of $A$ can be computed as the sum of each row(column) multiplied by the corresponding cofactor $det A = \sum ^n _{j=1} a_{i,j} A_{i,j}$ for any arbitrary $i$ $det A = \sum ^n _{i=1} a_{i,j} A_{i,j}$ for any arbitrary $j$

The determinant of a one-element matrix is just the element $(det(a = a))$, can compute the determinant of any square number

Theorem 2

Sum of the elements of a row(column) multiplied by the corresponding cofactor related to another row(column) is always zero $\sum ^n _{j=1} a_{i,j} A_{k,j}=0$ for any arbitrary $k\ne i$ $\sum ^n _{i=1} a_{i,j} A_{i,k}=0$ for any arbitrary $k\ne j$

minors are the row you remove, so you take sub matrix of the remaining part. $k$ = minor

Introduction to Matrix inversion

Inverting a matrix

For a square matrix, $A$, the inverse is $A^{-1}$ which is define as the matrix for which $AA^{-1}=I$

• Invertible Matrices: $\exists$ a matrix $B \in \R _{n,n} | AB=I=BA$
• Unique Inverse $\exists!$ a matrix $B \in \R _{n,n} | AB=I=BA$
• Inverse matrix $A^{-1}$: $A^{-1} = \frac{1}{detA} adj(A)$
• Singular/Non Invertable/ Linear Dependent: $det(A) =0$
• Non-singular: $det(A)\ne 0$
• Inverse of a matrix product: $(AB)^{-1}=B^{-1}A^{-1}$