Solve the linear equations by the matrix method:a+3b+2c=3,2a-b-3c=-8,5a+2b+c=9.solve for a:b:c

1 +3 +2 +3

2 -1 -3 -8
5 +2 +1 +9

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2, -3, 5

To solve the linear equations using the matrix method, we can represent the system of equations in matrix form.

Let's define the coefficient matrix A and the variable matrix X:

A = {{1, 3, 2},
{2, -1, -3},
{5, 2, 1}}

X = {{a},
{b},
{c}}

Also, define the constant matrix B:

B = {{3},
{-8},
{9}}

Now, the system of equations can be written in matrix form as AX = B.

To find the solution, we can use matrix operations to solve for X. Multiplying both sides of the equation by the inverse of A, we get:

X = A^(-1) * B

To find the inverse of matrix A, we proceed as follows:

1. Calculate the determinant of A:
det(A) = 1 * (-1 * 1 - 2 * -3) - 3 * (2 * 1 - 5 * -3) + 2 * (2 * -2 - 5 * -1)

2. If the determinant (det(A)) is non-zero, A^(-1) exists.

3. Calculate the adjugate of A:
adj(A) = {{-7, -11, 19},
{19, 7, -1},
{-5, -1, 1}}

4. Calculate the inverse of A:
A^(-1) = (1 / det(A)) * adj(A)

Now, we can calculate the inverse of A using these steps:

1. Calculate det(A):
det(A) = 1 * (1 - 6) - 3 * (2 + 15) + 2 * (-4 + 0)
= 1 * (-5) - 3 * 17 + 2 * (-4)
= -5 - 51 - 8
= -64

2. Calculate adj(A):
adj(A) = {{(-7), (-11), 19},
{19, 7, (-1)},
{(-5), (-1), 1}}

3. Calculate A^(-1):
A^(-1) = (1 / (-64)) * {{(-7), (-11), 19},
{19, 7, (-1)},
{(-5), (-1), 1}}
= {{1/64, 11/64, (-19/64)},
{(-19/64), (-7/64), 1/64},
{5/64, 1/64, 1/64}}

Finally, we can find the solution X by multiplying A^(-1) with B:

X = A^(-1) * B

Calculating X:

X = {{1/64, 11/64, (-19/64)},
{(-19/64), (-7/64), 1/64},
{5/64, 1/64, 1/64}}
*
{{3},
{-8},
{9}}

After performing the multiplication, we get:

X = {{1},
{2},
{1}}

Therefore, the solution to the system of equations is:
a = 1, b = 2, c = 1.