Determine whether the function involving the n × n matrix A is a linear transformation.T: Mn,n → Mn,n, T(A) = A−1
Question
Determine whether the function involving the n × n matrix A is a linear transformation.
Let the transformation be defined as:
Solution
To determine whether a function is a linear transformation, it must satisfy two properties:
- Additivity: T(A + B) = T(A) + T(B)
- Scalar multiplication: T(cA) = cT(A)
where A and B are matrices, and c is a scalar.
Let's check these properties for the function T(A) = A−1.
- Additivity:
T(A + B) = (A + B)−1 ≠ A−1 + B−1 = T(A) + T(B)
The additivity property is not satisfied.
- Scalar multiplication:
T(cA) = (cA)−1 ≠ cA−1 = cT(A)
The scalar multiplication property is not satisfied.
Therefore, the function T(A) = A−1 is not a linear transformation.
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