Does the matrix A =
2 1
5 4
have an inverse? If so, what is it?
To determine if the matrix A has an inverse, we need to calculate the determinant. The determinant of a 2x2 matrix A is given by:
det(A) = (a*d) - (b*c)
For matrix A =
2 1
5 4
det(A) = (2*4) - (1*5)
= 8 - 5
= 3
Since the determinant is non-zero (det(A) ≠ 0), matrix A has an inverse. The formula for the inverse of a 2x2 matrix A is:
A⁻¹ = (1/det(A)) * adj(A)
Where adj(A) represents the adjugate of matrix A, which is the transpose of the cofactor matrix. So, let's calculate:
Aᵀ =
2 5
1 4
Cofactor matrix of A =
4 -5
-1 2
adj(A) = Aᵀ (transpose of Aᵀ) =
2 1
5 4
Now, we can calculate the inverse matrix A⁻¹:
A⁻¹ = (1/det(A)) * adj(A)
= (1/3) *
2 1
5 4
=
2/3 1/3
5/3 4/3
Therefore, the inverse of matrix A is:
A⁻¹ =
2/3 1/3
5/3 4/3