Find the Nullity (2A^T)^-1=[[4,5],[3,4]]

Problem

Find the Nullity of *A* given *(2*A^T)^(−1)=[[4,5],[3,4]]

Solution

1. Invert the equation to solve for 2*A^T by taking the inverse of both sides.

((2*A^T)^(−1))^(−1)=[[4,5],[3,4]]^(−1)

2. Calculate the inverse of the 2×2 matrix using the formula 1/(a*d−b*c)*[[d,−b],[−c,a]]

2*A^T=1/((4)*(4)−(5)*(3))*[[4,−5],[−3,4]]

3. Simplify the scalar to find the expression for 2*A^T

2*A^T=1/(16−15)*[[4,−5],[−3,4]]

2*A^T=[[4,−5],[−3,4]]

4. Solve for A by dividing by 2 and then taking the transpose of the result.

A^T=[[2,−2.5],[−1.5,2]]

A=[[2,−1.5],[−2.5,2]]

5. Determine the rank of the matrix A Since the determinant of the original given matrix is 1≠0 the matrix (2*A^T)^(−1) is non-singular (invertible).

det((2*A^T)^(−1))=1

6. Apply properties of invertibility to conclude that if (2*A^T)^(−1) exists, then 2*A^T must be invertible, which implies A^T is invertible, and consequently A is invertible.

det(A)≠0

7. Use the Rank-Nullity Theorem which states that for an n×n matrix, rank(A)+nullity(A)=n For a 2×2 invertible matrix, the rank is 2

2+nullity(A)=2

8. Calculate the nullity by subtracting the rank from the number of columns.

nullity(A)=0

Final Answer

nullity(A)=0

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