If A is a square matrix, then the value of adj A T– (adj A) Tis equal to
null matrix whose order is same as that of A
Understanding the Matrix Problem: Adjoint and Transpose
The problem asks us to find the value of the expression \( \text{adj}(A^T) - (\text{adj } A)^T \) where \( A \) is a square matrix. To solve this, we need to recall and apply some fundamental properties of matrix operations, specifically the adjoint and the transpose of a matrix.
Key Matrix Properties
Let \( A \) and \( B \) be matrices of appropriate sizes, and \( I \) be the identity matrix. Here are some properties relevant to this problem:
- Transpose of a Transpose: \( (A^T)^T = A \)
- Transpose of a Product: \( (AB)^T = B^T A^T \)
- Property of Adjoint and Transpose: \( \text{adj}(A^T) = (\text{adj } A)^T \)
- Relationship between A, adj A, and Determinant: \( A (\text{adj } A) = (\text{adj } A) A = |A| I \)
- Adjoint of a Product: \( \text{adj}(AB) = (\text{adj } B)(\text{adj } A) \)
- Determinant of Adjoint: \( |\text{adj } A| = |A|^{n-1} \), where \( n \) is the order of the square matrix \( A \).
The third property listed, \( \text{adj}(A^T) = (\text{adj } A)^T \), is particularly important for solving this problem directly.
Evaluating the Expression
We are asked to find the value of the expression \( \text{adj}(A^T) - (\text{adj } A)^T \). Let's use the property \( \text{adj}(A^T) = (\text{adj } A)^T \).
Substitute \( \text{adj}(A^T) \) with \( (\text{adj } A)^T \) in the given expression:
\[ \text{adj}(A^T) - (\text{adj } A)^T \] \[ = (\text{adj } A)^T - (\text{adj } A)^T \]Subtracting a matrix from itself results in a zero matrix, also known as a null matrix.
So, the value of the expression is the null matrix.
Understanding the Null Matrix
A null matrix (or zero matrix) is a matrix where every element is zero. The order of the resulting null matrix will be the same as the order of the matrix \( A \), because \( A \) is a square matrix, \( A^T \) has the same order, \( \text{adj } A \) has the same order, and \( (\text{adj } A)^T \) has the same order.
Therefore, \( \text{adj}(A^T) - (\text{adj } A)^T \) is equal to the null matrix whose order is the same as that of \( A \).
Summary of the Solution Steps
- Identify the given expression: \( \text{adj}(A^T) - (\text{adj } A)^T \).
- Recall the property relating the adjoint of a transpose to the transpose of an adjoint: \( \text{adj}(A^T) = (\text{adj } A)^T \).
- Substitute this property into the expression: \( (\text{adj } A)^T - (\text{adj } A)^T \).
- Perform the subtraction: The result is the zero matrix (null matrix).
- Confirm the order: The resulting null matrix has the same order as the original square matrix \( A \).
This aligns with the option stating that the value is the null matrix whose order is the same as that of A.
Revision Table: Matrix Adjoint and Transpose
| Concept | Notation | Description/Property |
|---|---|---|
| Transpose | \( A^T \) | Matrix obtained by interchanging rows and columns of \( A \). \( (A^T)_{ij} = A_{ji} \). |
| Adjoint | \( \text{adj } A \) | Transpose of the cofactor matrix of \( A \). \( \text{adj } A = (C_{ij})^T \), where \( C_{ij} \) is the cofactor of \( a_{ij} \). |
| Adjoint-Transpose Property | \( \text{adj}(A^T) \) vs \( (\text{adj } A)^T \) | These two are equal: \( \text{adj}(A^T) = (\text{adj } A)^T \). |
| Null Matrix | \( \mathbf{0} \) | A matrix where all elements are zero. When subtracted from itself, a matrix results in the null matrix of the same order. |
Additional Information: Properties of Matrix Operations
Understanding the properties of matrix operations like addition, multiplication, transpose, determinant, and adjoint is crucial for solving matrix problems. Here are a few more general properties:
- Transpose of Sum: \( (A+B)^T = A^T + B^T \)
- Determinant of Transpose: \( |A^T| = |A| \)
- Adjoint of Inverse: \( \text{adj}(A^{-1}) = (\text{adj } A)^{-1} \) (if \( A \) is invertible)
- Inverse in terms of Adjoint: \( A^{-1} = \frac{1}{|A|} \text{adj } A \) (if \( |A| \neq 0 \))
These properties allow us to manipulate matrix expressions and simplify them to find their values or prove identities. The specific property \( \text{adj}(A^T) = (\text{adj } A)^T \) is a standard result derived from the definitions of adjoint and transpose, making the given problem straightforward once this property is known.
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