Table of contents
2 subforms (both are determinants)
2.2 k-order principal subforms
2.3 Sequential principal subforms of order k
3.2 Algebraic remainder expressions
3.3 What is the role of the remainder formula?
1 The original matrix A
- Next, design an original matrix A, deliberately designed as A34, the number of rows ≠ the number of columns
$$
\left[
\begin{matrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 &11 & 12 \\
\end{matrix}
\right]
$$
2 subforms (both are determinants)
- determinants
2.1 k-order subformula
- Randomly select k rows and k columns from a matrix, there will be k*k elements at the intersection, and these elements constitute the k-order determinant obtained by still maintaining the relative position order in the matrix, which is called the K-order subform of the matrix
- If a matrix Am*n if i∈m is a k-element subset, and j∈n is a k-element subset, then |A|i*j is the k-order subform of Am*n
- Simply put, the sub-formula is the determinant formed from the part of the matrix rotated in the matrix, the number of rows = the number of columns
For example, the first-order subform : because there is only 1 row and 1 column
$$
\left[
\begin{matrix}
1 \\
\end{matrix}
\right]
$$
$$
\left[
\begin{matrix}
7 \\
\end{matrix}
\right]
$$
For example, a 2-order subform : because there are 2 rows and 2 columns
$$
\left[
\begin{matrix}
1 & 2 \\
5 & 6 \\
\end{matrix}
\right]
$$
$$
\left[
\begin{matrix}
1 & 4 \\
5 & 8 \\
\end{matrix}
\right]
$$
For example, a 3-order subform: because there are 3 rows and 3 columns
$$
\left[
\begin{matrix}
1 & 3 & 4 \\
5 & 7 & 8 \\
9 &11 & 12 \\
\end{matrix}
\right]
$$
2.2 k-order principal subforms
- If the row numbers and column numbers are equal, it is the k-order principal subform
- If i=j, then |A|i*j is the k-order principal subform of Am*n
- Simply put, the main subform is the determinant formed from the part of the matrix rotated in the matrix , requiring the number of rows = the number of columns, and also requiring {row number array} = {column number array}
For example, the 1st-order principal subform: because there is 1 row and 1 column, and it is the 1st row and the 1st column
$$
\left[
\begin{matrix}
1 \\
\end{matrix}
\right]
$$
But the following sub-form is not the main sub-form, because the acquisition is the sub-form composed of the contents of the second row and the third column
$$
\left[
\begin{matrix}
7 \\
\end{matrix}
\right]
$$
For example, the 2nd-order principal subform: because there are 2 rows and 2 columns, and it is the 1st and 2nd rows, and the 1st and 2nd columns
$$
\left[
\begin{matrix}
1 & 2 \\
5 & 6 \\
\end{matrix}
\right]
$$
The following sub-form is still the main sub-form, because the acquisition is a sub-form composed of the contents of the 1st and 3rd rows and the 1st and 3rd columns
$$
\left[
\begin{matrix}
1 & 3 \\
9 & 11 \\
\end{matrix}
\right]
$$
But the following sub-form is not the main sub-form, because the acquisition is a sub-form composed of the contents of the 1st and 2nd rows and the 1st and 4th columns
$$
\left[
\begin{matrix}
1 & 4 \\
5 & 8 \\
\end{matrix}
\right]
$$
For example, the 3rd-order principal subform: because there are 3 rows and 3 columns, and it is the 1st, 2nd, 3rd rows, and the 1st, 2nd, 3rd columns
$$
\left[
\begin{matrix}
1 & 2 & 3 \\
5 & 6 & 7 \\
9 &10 & 11 \\
\end{matrix}
\right]
$$
But the following sub-form is not the main sub-form, because the acquisition is a sub-form composed of the contents of rows 1, 2 and 3 and columns 1, 3 and 4
$$
\left[
\begin{matrix}
1 & 3 & 4 \\
5 & 7 & 8 \\
9 &11 & 12 \\
\end{matrix}
\right]
$$
2.3 Sequential principal subforms of order k
- If i=j=(1,2....k), that is, the first k columns from the left and the first k rows from the top are obtained, then |A|i*j is the k-order order principal subform of Am*n
- To put it simply, the sequence main subform is the determinant formed by the part of the matrix rotated from the matrix . It requires the number of rows = the number of columns, and it also requires that {row number array} = {column number array}, and it must be according to the slave Take rows and columns in order from left to right, from top to bottom.
1st-order order principal subform: because there is 1 row and 1 column, and it is the 1st row and 1st column
$$
\left[
\begin{matrix}
1 \\
\end{matrix}
\right]
$$
2nd-order order principal subform: because there are 2 rows and 2 columns, and it is the 1st and 2nd rows, and the 1st and 2nd columns
$$
\left[
\begin{matrix}
1 & 2 \\
5 & 6 \\
\end{matrix}
\right]
$$
3rd-order order principal subform: because there are 3 rows and 3 columns, and it is the 1st, 2nd, 3rd rows, 1st, 2nd, 3rd columns
$$
\left[
\begin{matrix}
1 & 2 & 3 \\
5 & 6 & 7 \\
9 & 10 &11 \\
\end{matrix}
\right]
$$
3 remainder
The function is to simplify the n-order determinant to n-1 order determinant
3.1 Remainder formula
- In the nth-order determinant (meaning that it is a matrix corresponding to a square matrix), the content of the i-th row and j-th column where aij is located is crossed out, and the remaining determinant is called a remainder.
- Denote as Mij
3.2 Algebraic remainder expressions
strictly defined
- matrix, square matrix An*n
- The rest of the subtypes Mij
- The algebraic remainder formula is recorded as Cij=(-1)^(i+j)*Mij
3.3 What is the role of the remainder formula?
- The function is to simplify the n-order determinant to n-1 order determinant
- The transposed matrix of C is called the adjoint matrix of A. The adjoint matrix is similar to the inverse matrix and can be used to calculate its inverse matrix when A is invertible.
- The expansion of the 3rd-order determinant requires the calculation of the remainder
