Does the given matrix, A, have an inverse? If it does, what is A–1?
A=[-7 -25];[2 7]
To determine if matrix A has an inverse, we need to compute its determinant.
The determinant of matrix A is given by:
det(A) = (−7)(7) − (−25)(2)
Simplifying this expression, we get:
det(A) = -49 + 50 = 1
Since the determinant is nonzero (1 ≠ 0), matrix A does have an inverse.
Next, we can compute the inverse of matrix A using the formula:
A^-1 = (1/det(A)) * adj(A)
Where adj(A) represents the adjugate of matrix A.
The adjugate of matrix A is obtained by taking the transpose of its cofactor matrix.
The cofactor matrix of matrix A is obtained by taking the determinants of the minors of each element of A, alternating the signs accordingly.
Let's calculate the adjugate of A:
adj(A) = [7 -25; -2 -7]
Finally, we can compute the inverse of matrix A:
A^-1 = (1/1) * [7 -25; -2 -7]
Simplifying this expression, we get:
A^-1 = [7 -25; -2 -7]
To determine if a matrix has an inverse, we need to check if its determinant is non-zero.
The given matrix A is:
A = [-7 -25; 2 7]
To find the determinant of A, we use the formula:
det(A) = (a*d) - (b*c)
Where a, b, c, and d are the elements of the matrix.
For matrix A, the determinant is:
det(A) = (-7*7) - (-25*2)
= -49 + 50
= 1
Since the determinant of A is non-zero (1 ≠ 0), the matrix A does have an inverse.
To find the inverse of the matrix A, we can use the formula:
A^(-1) = (1/det(A)) * adj(A)
Where det(A) is the determinant of the matrix A and adj(A) is the adjoint of A.
First, let's find the adjoint of A by swapping the elements on the main diagonal and changing the sign of the elements on the off-diagonal:
adj(A) = [7 -25; -2 -7]
Then, multiply the adj(A) matrix by 1/det(A) to find the inverse of A:
A^(-1) = (1/1) * [7 -25; -2 -7]
= [7 -25; -2 -7]
Therefore, the inverse of the matrix A is:
A^(-1) = [7 -25; -2 -7]
To check whether a matrix has an inverse, we need to calculate its determinant.
The matrix A is given as:
A = [[-7, -25],
[2, 7]]
To find the determinant of A, we can use the following formula for a 2x2 matrix:
det(A) = (a * d) - (b * c)
In this case:
det(A) = (-7 * 7) - (-25 * 2)
= (-49) - (-50)
= -49 + 50
= 1
Since the determinant of A is not zero (det(A) ≠ 0), it means that A has an inverse.
To find the inverse of A, we can use the following formula for a 2x2 matrix:
A^-1 = (1/det(A)) * adj(A)
First, let's calculate the adjugate of A:
adj(A) = [[d, -b], [-c, a]]
= [[7, -(-25)], [-2, -7]]
= [[7, 25], [-2, -7]]
Now, let's calculate the inverse of A:
A^-1 = (1/det(A)) * adj(A)
= (1/1) * [[7, 25], [-2, -7]]
= [[7, 25], [-2, -7]]
So, the inverse of A is:
A^-1 = [[7, 25], [-2, -7]]