Use the inverse matrix to solve this system of equations:
4x+3y=7.5
7x+9z=14
4y-z=8.3
4,3,0
7,0,9
0,4,-1
To solve the system of equations using the inverse matrix, perform the following steps:
Step 1: Write the system of equations in matrix form.
⎡ 4 3 0 ⎤ ⎡ x ⎤ ⎡ 7.5 ⎤
⎢ ⎥ ⎢ ⎥ = ⎢ ⎥
⎢ 7 0 9 ⎥ ⎢ y ⎥ ⎢ 14 ⎥
⎢ ⎥ x ⎢ ⎥ ⎢ ⎥
⎣ 0 4 -1 ⎦ ⎣ z ⎦ ⎣ 8.3 ⎦
Step 2: Find the inverse of the coefficient matrix.
The coefficient matrix is:
⎡ 4 3 0 ⎤
⎢ ⎥
⎢ 7 0 9 ⎥
⎢ ⎥
⎣ 0 4 -1 ⎦
To find its inverse, perform row operations to convert it into an identity matrix while applying the same operations to the identity matrix as well:
⎡ 4 3 0 ⎤ ⎡ 1 0 0 ⎤
⎢ ⎥ ⎢ ⎥
⎢ 7 0 9 ⎥ → ⎢ 0 1 0 ⎥
⎢ ⎥ ⎢ ⎥
⎣ 0 4 -1 ⎦ ⎣ 0 0 1 ⎦
The resulting inverse matrix is:
⎡ -0.225 0.075 0.075 ⎤
⎢ ⎥
⎢ -0.175 0.175 -0.175 ⎥
⎢ ⎥
⎣ 0.2 -0.1 0.1 ⎦
Step 3: Multiply the inverse matrix with the constant matrix.
⎡ -0.225 0.075 0.075 ⎤ ⎡ 7.5 ⎤ ⎡ x ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ -0.175 0.175 -0.175 ⎥ × ⎢ 14 ⎥ = ⎢ y ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ 0.2 -0.1 0.1 ⎦ ⎣ 8.3 ⎦ ⎣ z ⎦
Performing the multiplication gives us:
⎡ x ⎤ ⎡ 1.3875 ⎤
⎢ ⎥ ⎢ ⎥
⎢ y ⎥ = ⎢ 18.3925⎥
⎢ ⎥ ⎢ ⎥
⎣ z ⎦ ⎣ 11.2775⎦
Therefore, the solution to the system of equations is:
x = 1.3875,
y = 18.3925,
z = 11.2775.
To solve the system of equations using the inverse matrix, follow these steps:
1. Write the system of equations in matrix form. Let's call the coefficient matrix A, the variable matrix X, and the constant matrix B.
A = | 4 3 0 |
| 7 0 9 |
| 0 4 -1 |
X = | x |
| y |
| z |
B = | 7.5 |
| 14 |
| 8.3 |
The equation AX = B represents the system in matrix form.
2. Find the inverse of matrix A, denoted as A^-1.
To find the inverse of A, apply the formula:
A^-1 = (1/det(A)) * adj(A)
Where det(A) is the determinant of A, and adj(A) is the adjugate of A.
3. Calculate the determinant of matrix A.
det(A) = (4 * 0 * -1) + (3 * 9 * 0) + (7 * 4 * 0) - (0 * 0 * 7) - (9 * 4 * -1) - (3 * 0 * 0)
= 0 + 0 + 0 - 0 + 36 + 0
= 36
4. Calculate the adjugate of matrix A.
The adjugate of matrix A is the transpose of the cofactor matrix. To find the cofactor matrix, calculate the determinant of each 2x2 submatrix and alternately multiply them by +1 or -1.
Cofactor matrix of A:
| (0 * -1) - (9 * 0) (7 * -1) - (0 * 0) (7 * 0) - (0 * 4) |
| (4 * -1) - (0 * 0) (4 * -1) - (7 * 0) (0 * 4) - (7 * 4) |
| (4 * 0) - (3 * 9) (0 * 0) - (4 * 0) (0 * 3) - (4 * 0) |
Cofactor matrix:
|-9 -7 0 |
|-4 -4 -28 |
|0 0 0 |
The adjugate of A is the transpose of the cofactor matrix:
Adj(A) = | -9 -4 0 |
| -7 -4 0 |
| 0 -28 0 |
5. Calculate the inverse of A.
A^-1 = (1/36) * adj(A)
A^-1 = | (-9/36) (-4/36) 0 |
| (-7/36) (-4/36) 0 |
| 0 (-28/36) 0 |
Simplify the coefficients to their decimal values:
A^-1 = | -0.25 -0.11 0 |
| -0.19 -0.11 0 |
| 0 -0.78 0 |
6. Multiply the inverse matrix A^-1 by B to solve for X.
X = A^-1 * B
X = | -0.25 -0.11 0 | | 7.5 |
| -0.19 -0.11 0 | * | 14 |
| 0 -0.78 0 | | 8.3 |
Perform the matrix multiplication:
X = | (-0.25 * 7.5) + (-0.11 * 14) + (0 * 8.3) |
| (-0.19 * 7.5) + (-0.11 * 14) + (0 * 8.3) |
| (0 * 7.5) + (-0.78 * 14) + (0 * 8.3) |
X = | -2.575 |
| -4.525 |
| -10.92 |
Therefore, the solution to the system of equations is:
x = -2.575
y = -4.525
z = -10.92