Does the given, A, If it does what is A^-1
A=[[-7,-25],[2,7]]
To find the inverse of matrix A, denoted as A^-1, we can use the formula for a 2x2 matrix given by:
Matrix A^-1 = (1 / determinant of A) * Adjugate of A
First, let's find the determinant of matrix A:
det(A) = (A11 * A22) - (A12 * A21)
= (-7 * 7) - (-25 * 2)
= -49 + 50
= 1
Next, let's find the cofactor matrix of A by swapping the values in the main diagonal and changing the signs of the values in the other diagonal:
Cofactor matrix of A = [[A22, -A21], [-A12, A11]]
= [[7, 25], [-2, -7]]
Now, we can find the adjugate matrix by transposing the cofactor matrix:
Adjugate matrix of A = [[A22, -A12], [A21, -A11]]
= [[7, -2], [25, -7]]
Finally, we can find the inverse of matrix A by dividing the adjugate matrix by the determinant of A:
A^-1 = (1 / determinant of A) * Adjugate of A
= (1 / 1) * [[7, -2], [25, -7]]
= [[7, -2], [25, -7]]
Therefore, the inverse of matrix A, A^-1, is [[7, -2], [25, -7]].
To find the inverse of matrix A, denoted as A^(-1), we can use the formula:
A^(-1) = (1/det(A)) * adj(A)
Where det(A) is the determinant of matrix A, and adj(A) is the adjugate of matrix A.
Now, let's calculate the inverse of matrix A step by step:
Step 1: Find the determinant of matrix A.
det(A) = (-7)*(7) - (-25)*(2) = -49 - (-50) = 1
Step 2: Find the adjugate of matrix A.
The adjugate of a 2x2 matrix can be obtained by switching the positions of the elements in the main diagonal and changing the sign of the elements in the off-diagonal positions.
For matrix A = [[-7, -25], [2, 7]]:
adj(A) = [[7, -2], [25, -7]]
Step 3: Calculate A^(-1).
A^(-1) = (1/det(A)) * adj(A)
= (1/1) * [[7, -2], [25, -7]]
= [[7, -2], [25, -7]].
Hence, the inverse of matrix A is:
A^(-1) = [[7, -2], [25, -7]].
To find the inverse of matrix A, denoted as A^-1, you can follow these steps:
Step 1: Write down the given matrix A:
A = [[-7, -25], [2, 7]]
Step 2: Calculate the determinant of matrix A using the formula:
det(A) = (a * d) - (b * c), where A = [[a, b], [c, d]]
a = -7, b = -25, c = 2, d = 7
det(A) = (-7 * 7) - (-25 * 2)
det(A) = (-49) - (-50)
det(A) = -49 + 50
det(A) = 1
Step 3: Check if A is invertible. A matrix is invertible if its determinant is non-zero. In this case, since det(A) = 1, the matrix A is invertible.
Step 4: Use the formula to find the inverse of matrix A:
A^-1 = (1/det(A)) * adj(A), where adj(A) represents the adjugate matrix of A.
Step 5: Calculate the adjugate matrix of A:
adj(A) = [[d, -b], [-c, a]], where A = [[a, b], [c, d]]
a = -7, b = -25, c = 2, d = 7
adj(A) = [[7, 25], [-2, -7]]
Step 6: Divide the adjugate matrix by the determinant:
A^-1 = (1/1) * [[7, 25], [-2, -7]]
A^-1 = [[7, 25], [-2, -7]]
Therefore, the inverse of matrix A is:
A^-1 = [[7, 25], [-2, -7]]