Does the given matrix, A, have an inverse? If it does, what is A–1?
A=[-7 -25];[2 7]
(Answer as a matrix)
To determine if a matrix has an inverse, we can calculate its determinant.
The given matrix is:
A = [-7 -25]
[ 2 7]
To calculate the determinant of matrix A, we use the formula:
det(A) = ad - bc
Where a, b, c, d are the elements of the matrix A. In our case:
det(A) = (-7)(7) - (-25)(2)
= -49 + 50
= 1
Since the determinant is not equal to zero (det(A) ≠ 0), the matrix A does have an inverse.
To find the inverse of matrix A, we use the formula:
A^(-1) = (1/det(A)) * adj(A)
Where adj(A) represents the adjugate of matrix A. The adjugate of a 2x2 matrix can be obtained by swapping the elements of the main diagonal and changing the sign of the elements in the off-diagonal. In our case:
adj(A) = [7 -25]
[-2 -7]
Now, we can calculate A^(-1):
A^(-1) = (1/1) * [7 -25]
[-2 -7]
= [7 -25]
[-2 -7]
Therefore, the inverse of matrix A is:
A^(-1) = [7 -25]
[-2 -7]
To determine if the given matrix A has an inverse, we can calculate its determinant. If the determinant is non-zero, then the matrix is invertible.
To find the determinant, we can use the formula: det(A) = ad - bc, where A = [a b; c d].
Let's calculate the determinant of matrix A:
A = [-7 -25; 2 7]
det(A) = (-7 * 7) - (-25 * 2)
= -49 + 50
= 1
Since the determinant of A is non-zero (1 ≠ 0), matrix A has an inverse. To find the inverse, we can use 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.
First, let's find the adjugate of A:
adj(A) = [d -b; -c a]
For matrix A, the adjugate is:
adj(A) = [7 25; -2 -7]
Next, let's calculate A⁻¹ using the formula:
A⁻¹ = (1/det(A)) * adj(A)
A⁻¹ = (1/1) * [7 25; -2 -7]
= [7 25; -2 -7]
Therefore, the inverse of matrix A, A⁻¹, is:
A⁻¹ = [7 25; -2 -7]
To determine if a matrix has an inverse, we need to check if its determinant is equal to zero. If the determinant is zero, then the matrix is said to be "singular" and does not have an inverse.
Let's find the determinant of matrix A:
Matrix A = [-7 -25; 2 7]
The determinant of a 2x2 matrix can be found using the formula:
det(A) = (a * d) - (b * c)
where a, b, c, and d are the elements of the matrix.
In this case:
a = -7
b = -25
c = 2
d = 7
det(A) = (-7 * 7) - (-25 * 2)
= (-49) - (-50)
= -49 + 50
= 1
Since the determinant of matrix A is equal to 1, it is non-zero. Therefore, matrix A has an inverse.
To find the inverse of matrix A, we can use the following formula for a 2x2 matrix:
A^(-1) = (1/det(A)) * adj(A)
where det(A) is the determinant of A, and adj(A) is the adjugate of A.
Let's calculate the inverse of matrix A:
Inverse of A = (1/det(A)) * adj(A)
The adjugate of a 2x2 matrix is obtained by swapping the positions of the elements along the main diagonal and changing the signs of the elements off the main diagonal.
adj(A) = [d -b; -c a]
= [7 -(-25); -2 -7]
= [7 25; -2 -7]
Inverse of A = (1/1) * adj(A)
= [7 25; -2 -7]
Thus, the inverse of matrix A is:
A^(-1) = [7 25; -2 -7]