Can someone explain to me step by step how the book got to this answer i want to understand the steps....thanks
Find the inverse, if it exists,for the matirx (*I used the underscore to separate the numbers within the matix)
-2_1_3
3_-1_2
-4_2_0
the answer is:
-4_6_5
-8_12_13
2_0_-1
The answer you supplied is really not the Inverse matrix since when you multiply the two you get
6 0 0
0 6 0
0 0 6 so each of the numbers in your answer should be divided by 6
There are several ways to find the inverse depending on your level of math knowledge.
Here is a rather straightforward way
let
│-2_1_3│ │a b c│ │1 0 0 │
│3_-1_2│ │d e f│=│0 1 0 │
│-4_2_0│ │g h i│ │0 0 1 │
from this we get
-2a + d + 3g = 1
3a - d + 2g = 0
-4a + 2d = 0
and
-2b + e + 3h = 0
3b - e + 2h = 1
-4b + 2e = 0
and
-2c + f + 3i = 0
3c - f + 2i = 0
-4c + 2f = 1
this gives you 3 systems of 3 equations in 3 variables to solve which do not appear to be that difficult.
Let me know what you got.
(Also check my math, I did not check it)
To find the inverse of a matrix, you can follow the steps provided in the explanation. Here's a step-by-step breakdown:
1. Start with the given matrix:
-2 1 3
3 -1 2
-4 2 0
2. Set up the matrix equation by adding the variables a, b, c, d, e, f, g, h, i, and the identity matrix:
-2 1 3 | 1 0 0
3 -1 2 | 0 1 0
-4 2 0 | 0 0 1
3. Establish the system of equations by equating the corresponding elements in the original matrix and the inverse matrix:
-2a + d + 3g = 1
3a - d + 2g = 0
-4a + 2d = 0
-2b + e + 3h = 0
3b - e + 2h = 1
-4b + 2e = 0
-2c + f + 3i = 0
3c - f + 2i = 0
-4c + 2f = 1
4. Solve the system of equations to find the values of a, b, c, d, e, f, g, h, i. You can use any method you prefer, such as substitution or matrix operations.
5. Once you have obtained the values of a, b, c, d, e, f, g, h, i, substitute them back into the matrix equation:
a b c | 1 0 0
d e f | 0 1 0
g h i | 0 0 1
6. Simplify the matrix by dividing each element by the determinant (in this case, 6), as mentioned in the provided answer:
-a/6 b/6 c/6 | 1/6 0 0
d/6 -e/6 f/6 | 0 1/6 0
g/6 h/6 -i/6 | 0 0 1/6
7. The resulting matrix is the inverse of the original matrix:
-a/6 b/6 c/6
d/6 -e/6 f/6
g/6 h/6 -i/6
Note that the values of a, b, c, d, e, f, g, h, i should be calculated using the previous step.
Finally, it is crucial to double-check the calculations to ensure accuracy.