If A is a square matrix, then what is adj (A -1 ) – (adj A) -1 equal to?
Null matrix
Analyzing the Matrix Expression: adj(A⁻¹) – (adj A)⁻¹
The question asks us to determine the value of the expression \( \text{adj}(A^{-1}) - (\text{adj } A)^{-1} \) where \( A \) is a square matrix.
For the inverse of a matrix \( A \), denoted by \( A^{-1} \), and the adjoint of a matrix \( A \), denoted by \( \text{adj } A \), to exist, we must assume that \( A \) is an invertible square matrix. This means that the determinant of \( A \), denoted by \( |A| \), is non-zero \( (|A| \neq 0) \).
Key Properties of Adjoint and Inverse Matrices
We will use the following fundamental properties of square matrices:
- For any invertible square matrix \( A \), \( A (\text{adj } A) = (\text{adj } A) A = |A| I \), where \( I \) is the identity matrix of the same order as \( A \).
- From the above property, if \( |A| \neq 0 \), we can write \( \text{adj } A = |A| A^{-1} \).
- The inverse of the inverse of a matrix is the original matrix itself, i.e., \( (A^{-1})^{-1} = A \).
- The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix, i.e., \( |A^{-1}| = \frac{1}{|A|} = |A|^{-1} \).
- For any invertible matrices \( B \) and \( C \), the inverse of their product is the product of their inverses in reverse order, i.e., \( (BC)^{-1} = C^{-1} B^{-1} \).
Deriving the Expression for \( (\text{adj } A)^{-1} \)
We know that \( \text{adj } A = |A| A^{-1} \) for an invertible matrix \( A \). Let's find the inverse of \( \text{adj } A \):
\[ (\text{adj } A)^{-1} = (|A| A^{-1})^{-1} \]
Using the property \( (cC)^{-1} = c^{-1} C^{-1} \) where \( c \) is a scalar and \( C \) is a matrix:
\[ (\text{adj } A)^{-1} = (|A|)^{-1} (A^{-1})^{-1} \]
Using the property \( (A^{-1})^{-1} = A \):
\[ (\text{adj } A)^{-1} = |A|^{-1} A \quad \text{ (Equation 1)} \]
Deriving the Expression for \( \text{adj}(A^{-1}) \)
Let \( B = A^{-1} \). Since \( A \) is invertible, \( A^{-1} \) is also invertible. We can use the property \( \text{adj } B = |B| B^{-1} \) for the matrix \( B \):
\[ \text{adj}(A^{-1}) = |A^{-1}| (A^{-1})^{-1} \]
Using the properties \( |A^{-1}| = |A|^{-1} \) and \( (A^{-1})^{-1} = A \):
\[ \text{adj}(A^{-1}) = |A|^{-1} A \quad \text{ (Equation 2)} \]
Evaluating \( \text{adj}(A^{-1}) – (\text{adj } A)^{-1} \)
Now we substitute the results from Equation 1 and Equation 2 into the given expression:
\[ \text{adj}(A^{-1}) - (\text{adj } A)^{-1} = (|A|^{-1} A) - (|A|^{-1} A) \]
Since both terms are identical, their difference is the null matrix (a matrix of the same order as \( A \) with all its elements being zero).
\[ \text{adj}(A^{-1}) - (\text{adj } A)^{-1} = \mathbf{0} \]
Where \( \mathbf{0} \) represents the null matrix.
Comparing with Options
Let's look at the given options:
- \( 2|A| \)
- Null matrix
- Unit matrix
- None of the above
Our derived result is the null matrix.
| Expression Part | Derived Value |
|---|---|
| \( \text{adj}(A^{-1}) \) | \( |A|^{-1} A \) |
| \( (\text{adj } A)^{-1} \) | \( |A|^{-1} A \) |
| Difference | \( |A|^{-1} A - |A|^{-1} A = \mathbf{0} \) (Null Matrix) |
Therefore, \( \text{adj}(A^{-1}) – (\text{adj } A)^{-1} \) is equal to the Null matrix.
Revision Table: Adjoint and Inverse Matrix Properties
| Property | Description | Condition |
|---|---|---|
| \( A (\text{adj } A) = |A| I \) | Product of matrix and its adjoint | For any square matrix \( A \) |
| \( \text{adj } A = |A| A^{-1} \) | Adjoint in terms of inverse | For invertible matrix \( A \) \( (|A| \neq 0) \) |
| \( |A^{-1}| = |A|^{-1} \) | Determinant of inverse | For invertible matrix \( A \) \( (|A| \neq 0) \) |
| \( (A^{-1})^{-1} = A \) | Inverse of inverse | For invertible matrix \( A \) \( (|A| \neq 0) \) |
| \( (\text{adj } A)^{-1} = |A|^{-1} A \) | Inverse of adjoint | For invertible matrix \( A \) \( (|A| \neq 0) \) |
| \( \text{adj}(A^{-1}) = |A|^{-1} A \) | Adjoint of inverse | For invertible matrix \( A \) \( (|A| \neq 0) \) |
Additional Information on Null Matrix and Unit Matrix
The Null Matrix (or Zero Matrix) is a matrix where all its elements are zero. It is usually denoted by \( \mathbf{0} \). For square matrices of the same order, adding or subtracting the null matrix does not change the matrix. For example, \( M + \mathbf{0} = M \) and \( M - \mathbf{0} = M \).
The Unit Matrix (or Identity Matrix) is a square matrix with ones on the main diagonal and zeros elsewhere. It is usually denoted by \( I \) or \( I_n \) (where \( n \) is the order of the matrix). It acts as the multiplicative identity for matrices, i.e., \( M I = I M = M \).
In this problem, the result of the expression \( \text{adj}(A^{-1}) - (\text{adj } A)^{-1} \) is the Null matrix, meaning it is a matrix filled entirely with zeros.
Important Questions from Adjoint and Inverse of a Square Matrix
Let A be a matrix of order 3 × 3 and |A| = 4. If |2adj(3A)| = 2α 3β, then what is the value of (α + β)?
The matrix \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&3&2\\ 1&{{\rm{x}} - 1}&1\\ 2&7&{{\rm{x}} - 3} \end{array}} \right]\)
Will have inverse for every real number x except forIf $A = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix}$, then what is $A + 3A^{-1}$ equal to, where $A$ is a matrix of order 2?
Find the value of $|adj (2 \cdot adj A)|$ if matrix $A$ is of the order of $3$ and $|A| = 15$.
- If $A$ is a singular matrix of order $n \ge 2$, then which of the following statements is true?
