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Introduction: because of C language to get acquainted with programming, then transferred to the computer major, had the honor to get some national awards, provincial awards… Has been confirmed. Currently learning C++/Linux/Python

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1.7 Cramer’s Rule

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With n unknowns x1,x2… ,xnx_1,x_2,… ,x_nx1,x2,… Xn system of n linear equations


{ a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 . . . . . . . . . . . . . . . . . . . . a n 1 x 1 + a n 2 x 2 + . . . + a n n x n = b n (1) \begin{cases} a_{11}x_1+a_{12}x_2+… +a_{1n}x_n=b_1 \\ a_{21}x_1+a_{22}x_2+… +a_{2n}x_n=b_2 \\ ……………….. \\ a_{n1}x_1+a_{n2}x_2+… +a_{nn}x_n=b_n \\ \end{cases} \tag1

Kramer’s law: If the determinant of the coefficients of the system of linear equations (1) is not equal to zero, i.e


D = ∣ a 11 . . . a 1 n a 21 . . . a 2 n . . . . a n 1 . . . a n n ∣ indicates 0 D=\begin{vmatrix} a_{11} &… & a_{1n}\\ a_{21} & … &a_{2n}\\ . & &. \\ . & & . \\ a_{n1} &… & a_{nn}\\ \end{vmatrix} \neq 0

Then, the system (1) has a unique solution


x 1 = D 1 D . x 2 = D 2 D . . . . . x n = D n D x_1=\frac{D_1}{D},x_2=\frac{D_2}{D},… ,x_n=\frac{D_n}{D}

Among them, the Dj (j = 1, 2,… , n) D_j (j = 1, 2,… , n) Dj (j = 1, 2,… ,n) is the n-order determinant obtained by substituting the JTH column element in the coefficient determinant DDD with the constant term at the right end of the equations.

namely


D j = ∣ a 11 . . a 1 . j – 1 . . b 1 a 1 . j + 1 . . a 1 n . . . . . . . . . . . . . . . a n 1 . . a n . j – 1 . . b n a n . j + 1 . . a n n ∣ D_j=\begin{vmatrix} a_{11} &.. & a_{1,j-1}&.. &b_1&a_{1,j+1}&.. & a_{1n}\\ . & &. & & . & .& &. \\ . & &. & & . & .& &. \\ . & &. & & . & .& &. \\ a_{n1} &.. & a_{n,j-1}&.. &b_n&a_{n,j+1}&.. & a_{nn}\\ \end{vmatrix}

Theorem 4

If the determinant of the coefficients of the linear equations (1) is D≠0D \neq 0D=0, then (1) must have a solution and the solution is unique

Inverse negation of theorem 4

If the system of linear equations (1) has no solution or two different solutions, the determinant of its coefficients must be 0

Nonsingular/singular linear equations

Odd degree linear system of equations


{ a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = 0 . . . . . . . . . . . . . . . . . . . . a n 1 x 1 + a n 2 x 2 + . . . + a n n x n = 0 \begin{cases} a_{11}x_1+a_{12}x_2+… +a_{1n}x_n=0 \\ a_{21}x_1+a_{22}x_2+… +a_{2n}x_n=0 \\ ……………….. \\ a_{n1}x_1+a_{n2}x_2+… +a_{nn}x_n=0 \\ \end{cases}

Nonsingular linear equations (B1, B2… Bnb_1 b_2… B_nb1, b2… Bn is not all zero.)


{ a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = b 2 . . . . . . . . . . . . . . . . . . . . a n 1 x 1 + a n 2 x 2 + . . . + a n n x n = b n \begin{cases} a_{11}x_1+a_{12}x_2+… +a_{1n}x_n=b_1 \\ a_{21}x_1+a_{22}x_2+… +a_{2n}x_n=b_2 \\ ……………….. \\ a_{n1}x_1+a_{n2}x_2+… +a_{nn}x_n=b_n \\ \end{cases}

For singular linear equations


{ a 11 x 1 + a 12 x 2 + . . . + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + . . . + a 2 n x n = 0 . . . . . . . . . . . . . . . . . . . . a n 1 x 1 + a n 2 x 2 + . . . + a n n x n = 0 \begin{cases} a_{11}x_1+a_{12}x_2+… +a_{1n}x_n=0 \\ a_{21}x_1+a_{22}x_2+… +a_{2n}x_n=0 \\ ……………….. \\ a_{n1}x_1+a_{n2}x_2+… +a_{nn}x_n=0 \\ \end{cases}

x1=x2=… =xn=0x_1=x_2=… =x_n=0x1=x2=… Xn equals 0 must be the solution, and the solution is called the zero solution. A set of solutions that are not all zero is said to be nonzero. Singular linear equations must have zero solutions, but not necessarily non-zero solutions.

Theorem 5

If the coefficient determinant of the system of odd linear equations D≠0D \neq0D=0, the system of odd linear equations has no non-zero solution.

Because D≠0D \neq0D=0, it means that there is only one solution, and the zero solution must exist, so the solution can only be zero. If the system of odd linear equations has a non-zero solution, then the determinant of its coefficients D=0D =0D=0

conclusion

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