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:

1. \( 2|A| \)
2. Null matrix
3. Unit matrix
4. None of the above

Our derived result is the null matrix.

Therefore, \( \text{adj}(A^{-1}) – (\text{adj } A)^{-1} \) is equal to the Null matrix.

Revision Table: Adjoint and Inverse Matrix Properties

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.