Question 1196276: If m=(1 -1 K)
4 7 3
-1 12 -2
Evaluate in term of K the determinant of m
Hence,if x=(x)
Y
Z
Solve the equation mx=(1)
11
21
When K=Z,by the lot method
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! Certainly, let's evaluate the determinant of matrix 'm' and solve the equation 'mx = (1, 11, 21)' when K = Z.
**1. Evaluate the Determinant of Matrix 'm'**
* **Expand along the first row:**
det(m) = 1 * det(
7 3
12 -2
)
- (-1) * det(
4 3
-1 -2
)
+ K * det(
4 7
-1 12
)
det(m) = 1 * (7 * (-2) - 3 * 12) + 1 * (4 * (-2) - 3 * (-1)) + K * (4 * 12 - 7 * (-1))
det(m) = 1 * (-14 - 36) + 1 * (-8 + 3) + K * (48 + 7)
det(m) = -50 - 5 + 55K
det(m) = 55K - 55
**2. Solve the Equation 'mx = (1, 11, 21)' when K = Z**
* **Substitute K = Z into matrix 'm':**
m = (
1 -1 Z
4 7 3
-1 12 -2
)
* **Set up the augmented matrix:**
[
1 -1 Z | 1
4 7 3 | 11
-1 12 -2 | 21
]
* **Perform row operations to get the matrix in row-echelon form:**
1. **R2 = R2 - 4R1**
[
1 -1 Z | 1
0 11 3-4Z | 7
-1 12 -2 | 21
]
2. **R3 = R3 + R1**
[
1 -1 Z | 1
0 11 3-4Z | 7
0 11 Z-2 | 22
]
3. **R3 = R3 - R2**
[
1 -1 Z | 1
0 11 3-4Z | 7
0 0 Z+1 | 15
]
* **Solve for Z:**
* Z + 1 = 15
* Z = 14
* **Back-substitute to find Y:**
* 11Y + 3 - 4Z = 7
* 11Y + 3 - 4(14) = 7
* 11Y - 53 = 7
* 11Y = 60
* Y = 60/11
* **Back-substitute to find X:**
* X - Y + Z = 1
* X - (60/11) + 14 = 1
* X = 1 + (60/11) - 14
* X = -3 + (60/11)
* X = 27/11
**Therefore, when K = Z, the solution to the equation 'mx = (1, 11, 21)' is:**
* **X = 27/11**
* **Y = 60/11**
* **Z = 14**
I hope this comprehensive solution is helpful!
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