If det(A)=-7
A= | a b c |
| d e f |
| g h i |
Find det(2A^-1), det((2A)^-1).
Find
| a g d |
| b h e |
| c i f |
As you can see from your own post, it is very difficult to type and show matirices and determinants in this format.
Take a look at these sites, thy summarize most of the important properties very nicely.
http://www.sosmath.com/matrix/determ0/determ0.html
http://www.ping.be/~ping1339/determ.htm
To find the determinant of a given matrix A, we need to plug in the given values into the determinant formula. So, let's find the determinants step-by-step.
Given: det(A) = -7
A = | a b c |
| d e f |
| g h i |
Determinant of a 3x3 matrix A is given by the formula:
det(A) = a(ei - fh) - b(di - fg) + c(dh - ge)
Step 1: Calculate the determinants
det(A) = a(ei - fh) - b(di - fg) + c(dh - ge)
= a(ei - fh) - b(di - fg) + c(dh - ge)
We need to solve this equation to get the value of determinant A.
Step 2: Calculate the inverse of matrix A
To find the inverse of matrix A, we need to use the following formula:
A^(-1) = (1/det(A)) * adj(A)
Where adj(A) is the adjugate of matrix A, which is obtained by swapping the main diagonal of A and changing the sign of the off-diagonal elements.
Step 3: Calculate det(2A^(-1))
det(2A^(-1)) = 2^3 * det(A^(-1))
= 8 * det(A^(-1))
Step 4: Calculate det((2A)^(-1))
det((2A)^(-1)) = (2^3) * det(A)^(-1)
= 8 / det(A)
Step 5: Calculate the matrix
To find the new matrix, we just need to arrange the given elements in the required order.
New matrix = | a g d |
| b h e |
| c i f |
Hope this helps! Let me know if you have any more questions.
To find det(2A^-1), we need to determine the inverse of matrix A first.
Step 1: Calculate the determinant of matrix A: det(A) = -7.
Step 2: To find the inverse of matrix A (A^-1), you can use the formula:
A^-1 = (1/det(A)) * adj(A)
where adj(A) denotes the adjugate of matrix A.
Step 3: Calculate the adjugate of matrix A by finding the transpose of the matrix of cofactors of A.
The matrix of cofactors of A can be obtained by taking the determinant of each 2x2 submatrix of A with alternating signs.
Step 4: Divide each element of the matrix of cofactors by the determinant of A to get the adjugate.
Let's perform these calculations step by step:
Step 1: Calculate det(A) = -7.
Step 2: Find the inverse of matrix A (A^-1):
A^-1 = (1/det(A)) * adj(A)
Step 3: First, let's calculate the matrix of cofactors for matrix A:
Cofactor of a = det(| e f |) = e
| h i |
Cofactor of b = -det(| d f |) = -(d * i - f * g)
| g i |
Cofactor of c = det(| d e |) = d
| g h |
Cofactor of d = -det(| b c |) = -(b * i - c * h)
| h i |
Cofactor of e = det(| a c |) = a
| g i |
Cofactor of f = -det(| a b |) = -(a * i - b * g)
| d g |
Cofactor of g = det(| b c |) = b
| e f |
Cofactor of h = -det(| a c |) = -(a * f - c * d)
| e f |
Cofactor of i = det(| a b |) = a
| d e |
Step 4: Now, we can calculate the adjugate of matrix A by taking the transpose of the matrix of cofactors:
adj(A) = | e -(d * i - f * g) d |
| a -(a * i - b * g) a |
| b -(a * f - c * d) a |
Step 5: Divide each element of the adjugate matrix by the determinant of A:
A^-1 = (1/-7) * | e -(d * i - f * g) d |
| a -(a * i - b * g) a |
| b -(a * f - c * d) a |
Now that we have the inverse of matrix A, we can calculate the determinants of the given expressions.
To find det(2A^-1):
We simply multiply the determinant of A^-1 by 2.
det(2A^-1) = 2 * det(A^-1)
To find det((2A)^-1):
We first need to calculate 2A and then find its inverse.
2A = 2 * | a b c |
| d e f |
| g h i |
Then we calculate the inverse of 2A:
(2A)^-1 = (1/det(2A)) * adj(2A)
Finally, we can determine the determinants using the above formulas.