if a is a square matrix of order 5 adj a 81
If (A) is a square matrix of order 5, and (\text{adj}(A) = 81):
Understanding the Problem
The given information is about a square matrix (A) of order 5 (i.e., (A) is a (5 \times 5) matrix). It’s also mentioned that (\text{adj}(A) = 81). This information refers to the adjugate (also called the adjoint) matrix of (A). However, a value like (81) being used for (\text{adj}(A)) seems unusual because:
- The adjugate of (A), denoted by (\text{adj}(A)), is itself another matrix, not a scalar. Hence, (\text{adj}(A)) cannot directly represent a single number like (81).
- Based on matrix theory, the adjugate matrix is closely related to the determinant of (A).
Key Concepts to Clarify
Let’s express the relationship between a matrix, its adjugate, and its determinant:
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Adjugate Matrix Definition: The adjugate of a matrix (A), denoted (\text{adj}(A)), is the transpose of the cofactor matrix of (A).
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Property of Adjugate Matrix:
For an (n \times n) matrix (A), the property ties adjugate and determinant as:A \cdot \text{adj}(A) = \det(A) \cdot I_nwhere (I_n) is the (n \times n) identity matrix. The determinant of (A) is denoted (\det(A)).
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Special Case for Determinant:
If (A) is non-singular (i.e., (\det(A) \neq 0)), we can use the formula:\text{adj}(A) = (\det(A))^{n-1} A^{-1}.
Interpreting the Given Value
When the problem says (\text{adj}(A) = 81), it is likely referring to:
- The Determinant of the Adjugate Matrix: (\det(\text{adj}(A)) = 81),
or - Scalar Information Foreshadowing (\det(A)): This may indirectly imply a special property about determinant values.
If (\det(A)) is to be Determined
If they mean that (\det(\text{adj}(A)) = 81), then we use the following property:
For a (5 \times 5) matrix ((n = 5)):
Thus:
To solve for (\det(A)):
- Take the fourth root of 81:\det(A) = \pm \sqrt[4]{81}.
- Simplify:\sqrt[4]{81} = \sqrt{\sqrt{81}} = \sqrt{9} = 3.Therefore:\det(A) = \pm 3.
Final Answer
If (\text{adj}(A) = 81) refers to (\det(\text{adj}(A))):
Let me know if there’s any confusion! @username