Solve the set of linear equations by the matrix method. a+3b+2c= 3, 2a-b-3c= -8, 5a+2b+c=9, solve for a
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To solve the given set of linear equations using the matrix method, we need to represent the equations in matrix form and apply matrix operations. Here's how we can do it step by step:
Step 1: Rewrite the equations in matrix form:
We can write the set of equations as AX = B, where A represents the coefficient matrix, X represents the variable matrix, and B represents the constant matrix.
The equations can be rewritten as:
1a + 3b + 2c = 3 -- Equation 1
2a - 1b - 3c = -8 -- Equation 2
5a + 2b + 1c = 9 -- Equation 3
In matrix form, it can be written as:
[1 3 2] [a] [ 3 ]
[2 -1 -3] * [b] = [-8 ]
[5 2 1] [c] [ 9 ]
Step 2: Create the coefficient matrix (A):
The coefficient matrix (A) is formed by extracting the coefficients of the variables from the set of equations. In this case, the coefficient matrix (A) is:
[1 3 2]
[2 -1 -3]
[5 2 1]
Step 3: Create the variable matrix (X):
The variable matrix (X) represents the variables (a, b, c) we want to solve. In this case, the variable matrix (X) is:
[a]
[b]
[c]
Step 4: Create the constant matrix (B):
The constant matrix (B) represents the constants on the right-hand side of the equations. In this case, the constant matrix (B) is:
[ 3 ]
[-8 ]
[ 9 ]
Step 5: Find the inverse of coefficient matrix (A):
To solve for the variable matrix (X), we need to find the inverse of the coefficient matrix (A). We can represent the inverse of A as A^(-1).
Step 6: Calculate the variable matrix (X):
To find the values of the variables (a, b, c), we can multiply the inverse of the coefficient matrix (A^(-1)) by the constant matrix (B).
X = A^(-1) * B
Once we perform this matrix multiplication, we will get the values for a, b, and c, where a represents the first element, b represents the second element, and c represents the third element of the variable matrix (X).
Now, you can use any appropriate method to find the inverse of the matrix A, such as the Gauss-Jordan elimination method or using matrix calculators/formulas. After finding the inverse, multiply it with the constant matrix (B) to solve for the variable matrix (X).