step by step how do we find the value of x and y in this matrices by the law of matrices (3 2 4 0) (x y)=(7 12)
To find the values of x and y in the given matrices equation using the law of matrices, you need to multiply the inverse of the coefficient matrix (the first matrix in the equation) by the vector on the right-hand side (second matrix in the equation).
Let's go step by step:
1. Start by rearranging the equation to isolate the vector (x, y) on the left-hand side. Your equation becomes:
(3 2) (x) = (7)
(4 0) (y) (12)
2. Compute the inverse of the coefficient matrix (3 2 / 4 0). To find the inverse, you need to calculate the determinant of the matrix first. The determinant can be found by multiplying the top left element (3) by the bottom right element (0) and subtracting the product of the top right (2) and bottom left (4) elements.
Determinant (D) = (3 * 0) - (2 * 4) = 0 - 8 = -8
Since the determinant is not zero, the matrix is invertible.
Now, find the adjugate of the matrix, which is the transpose of the matrix of cofactors.
Adjugate = (0 -2) = (0 -2)
(-4 3) (-4 3)
Finally, multiply the adjugate by the reciprocal of the determinant to find the inverse of the coefficient matrix.
Inverse = (1/D) * Adjugate
= (1/-8) * (0 -2) = (0 1/4)
(-4 3/8)
3. Multiply the inverse of the coefficient matrix by the vector on the right-hand side (7 12) to solve for (x, y):
(0 1/4) (x) = (7)
(-4 3/8) (y) (12)
This can be expressed as:
0 * x + (1/4) * y = 7 --> Equation 1
-4 * x + (3/8) * y = 12 --> Equation 2
4. Solve the system of equations from step 3 to find the values of x and y. You can use various methods like substitution, elimination, or matrix operations.
Let's solve using the elimination method:
Multiply Equation 1 by 4 to eliminate the fraction:
0 * x + (4/4) * y = 28 --> Equation 1'
-4 * x + (3/8) * y = 12 --> Equation 2
Now, add Equation 1' and Equation 2:
-4 * x + (3/8) * y + 0 * x + (4/4) * y = 12 + 28
-4 * x + (7/8) * y = 40 --> Equation 3
Multiply Equation 3 by 8 to eliminate the fraction:
-32 * x + 7 * y = 320 --> Equation 3'
Now, multiply Equation 2 by 8 to eliminate the fraction:
-32 * x + 6 * y = 96 --> Equation 4
Subtract Equation 4 from Equation 3':
(-32 * x - (-32 * x)) + (7 * y - 6 * y) = 320 - 96
0 * x + y = 224
Simplify:
y = 224
Substitute the value of y back into Equation 1 to solve for x:
(1/4) * 224 = 7
56 = 7
There is no value of x that will satisfy this equation since 56 does not equal 7. Therefore, there is no solution to the given system of equations.
In conclusion, the system of equations does not have a solution for x and y.