If a is a square matrix of order 5 adj a 81

General
1

if a is a square matrix of order 5 adj a 81

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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:

  1. 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).
  2. 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:

  1. Adjugate Matrix Definition: The adjugate of a matrix (A), denoted (\text{adj}(A)), is the transpose of the cofactor matrix of (A).

  2. 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_n

    where (I_n) is the (n \times n) identity matrix. The determinant of (A) is denoted (\det(A)).

  3. 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:

  1. The Determinant of the Adjugate Matrix: (\det(\text{adj}(A)) = 81),
    or
  2. 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:

\det(\text{adj}(A)) = (\det(A))^{n-1}.

For a (5 \times 5) matrix ((n = 5)):

\det(\text{adj}(A)) = (\det(A))^{5-1} = (\det(A))^4.

Thus:

(\det(A))^4 = 81.

To solve for (\det(A)):

  1. Take the fourth root of 81:
    \det(A) = \pm \sqrt[4]{81}.
  2. 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))):

\det(A) = \pm 3.

Let me know if there’s any confusion! @username