Use an inverse matrix to solve each system of linear equations.
A) x+2y=-1 B)x+2y=10 C)x+2y=-3
x-2y=3 x-2y=-6 x-2y=0
To solve each system of linear equations using an inverse matrix, we can express the system in matrix form, AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Let's solve each system of linear equations step by step:
A) x + 2y = -1
x - 2y = 3
In matrix form, the system can be written as:
| 1 2 | | x | | -1 |
| 1 -2 | * | y | = | 3 |
To find the inverse of the coefficient matrix, we need to calculate the determinant of A. Let's first calculate the determinant:
det(A) = (1 * -2) - (2 * 1) = -4
Since the determinant is not equal to zero, A is invertible (non-singular).
Next, we can find the inverse matrix (A^(-1)) using the formula:
A^(-1) = (1/det(A)) * adj(A)
adj(A) is the adjugate matrix of A, which is the transpose of the cofactor matrix.
First, let's find the transpose of A:
A^T = | 1 1 |
| 2 -2 |
Next, we need to find the cofactor matrix, C, which is obtained by taking the determinants of the 2x2 matrices formed by eliminating each element from A^T. The cofactor Cij of each element aij of A^T is given by:
Cij = (-1)^(i+j) * det(Mij)
Where Mij is the matrix obtained by eliminating the ith row and jth column from A^T.
For our matrix, we have:
C11 = (-1)^(1+1) * det(M11) = (-1) * (-2) = 2
C12 = (-1)^(1+2) * det(M12) = (1) * (-2) = -2
C21 = (-1)^(2+1) * det(M21) = (1) * 2 = 2
C22 = (-1)^(2+2) * det(M22) = (-1) * 1 = -1
Now, let's form the cofactor matrix:
C = | 2 -2 |
| 2 -1 |
Finally, we can find the inverse of A using the formula:
A^(-1) = (1/det(A)) * adj(A) = (1/-4) * C = (1/-4) * | 2 -2 | = | -1/2 1/2 |
| -1/2 1/4 |
Now, we can find the variable matrix X by multiplying the inverse of A with the constant matrix B:
X = A^(-1) * B = | -1/2 1/2 | * | -1 | = | -2 |
| 1/4 | | 1 |
Therefore, the solution to the system of linear equations A) x + 2y = -1 and x - 2y = 3 is x = -2 and y = 1.
You can apply the same steps to solve systems B) and C) using the given equations and coefficients.