Linear algebra solves this problem! Cramer's law Find the cubic polynomial f (x) = A0 + a1x + a2x ^ 2 + a3x ^ 3, such that f (- 1) = 0, f (1) = 4, f (2) = 3, f (3) = 16

According to the meaning of the question, the following equation is obtained
a0-a1+a2-a3=0
a0+a1+a2+a3=4
a0+2a1+4a2+8a3=3
a0+3a1+9a2+27a3=16
Determinant of available coefficient
D= 1 -1 1 -1
1 1 1 1
1 2 4 8
1 3 9 27
D = 48, so D is not equal to 0
Therefore, the Cramer rule can be used
D1=0 -1 1 -1
4 1 1 1
3 2 4 8
16 3 9 27
D2=1 0 1 -1
1 4 1 1
1 3 4 8
1 16 9 27
D3=1 -1 0 -1
1 1 4 1
1 2 3 8
1 3 16 27
D4=1 -1 1 0
1 1 1 4
1 2 4 3
1 3 9 16
a0=D1/D=336/48=7
a1=D2/D=-132/48=-11/4
a2=D3/D=-240/48=5
a3=D4/D=96/48=2

Use Cramer's law to solve the following equations

X = 29 / 48y = 47 / 48Z = - 11 / 48 If there are n unknowns, the system of equations composed of n equations: Cramer's law a11x1 + a12x2 +... + a1nxn = B1, a21x1 + a22x2 +... + a2nxn = B2,... An1x1 + an2x2 +... + annxn = BN

How does D of Cramer's law work out 27?

Here I put forward my understanding of linear algebra. In addition to Cramer's rule, the most commonly used method to find the solution of linear equations is the elementary transformation method. It is very convenient to find the solution after transforming the corresponding augmented matrix of equations into the simplest form of rows. Cramer's rule can be used for binary or ternary equations, but not for equations with more than four variables, Because if we want to use the equation of four or more variables, we need to find the fourth order determinant or higher order determinant. If there is no simplification, it is very difficult to calculate by hand

Is Cramer's law a sufficient and necessary condition?
2. Is the converse no theorem of Cramer's law a sufficient and necessary condition?
3. If it is a sufficient and necessary condition, why not write it in the book? Does the law mean that it is a sufficient and necessary condition?

1 yes
2 yes
The conclusion can be obtained by analyzing and proving the process

Linear algebra solves this problem! Cramer's law Find the cubic polynomial f (x) = A0 + a1x + a2x ^ 2 + a3x ^ 3, such that f (- 1) = 0, f (1) = 4, f (2) = 3, f (3) = 16