Question

does the given matrix A have an inverse if it does what is A^-1
A=[-7 -25 2 7]

Answer 1

In order to determine if a matrix has an inverse, we need to find its determinant. If the determinant is zero, then the matrix does not have an inverse.

Let's compute the determinant of the matrix A:

A = [-7 -25 2 7]

The determinant of A is given by:

det(A) = (-7)(-25) - (2)(7)

det(A) = 175 - 14

det(A) = 161

Since the determinant of A is not zero, we can conclude that matrix A has an inverse.

To find A^(-1), we can use the formula:

A^(-1) = (1/det(A)) * adj(A)

Where * represents matrix multiplication, det(A) is the determinant of A, and adj(A) is the adjugate of A.

Since A is a 1x4 matrix, its adjugate will be a 4x1 matrix. Let's calculate the adjugate:

adj(A) = [-25]
[ -7]
[ 7]
[ 2]

Now, we can calculate A^(-1):

A^(-1) = (1/161) * adj(A)

Multiplying each element of adj(A) by (1/161), we get:

A^(-1) = [-25/161]
[ -7/161]
[ 7/161]
[ 2/161]

Therefore, the inverse of matrix A is:

A^(-1) = [-25/161]
[ -7/161]
[ 7/161]
[ 2/161]