Question

Let \( A = \begin{pmatrix} m & n \\ p & q \end{pmatrix} \), \( d = |A| \neq 0 \), and \( |A - d(\text{Adj}(A))| = 0 \). Then

• A. \(1 + d^2 = (m + q)^2\)
• B. \((1 + d)^2 = m^2 + q^2\)
• C. \((1 + d)^2 = (m + q)^2\)
• D. \(1 + d^2 = m^2 + q^2\)

Answer 1

The Correct Option is C

The given matrix is:

\[ A = \begin{bmatrix} m & n \\ p & q \end{bmatrix}. \]

Determinant of \( A \):

\[ |A| = m \cdot q - n \cdot p = d. \]

Adjugate of \( A \):

The adjugate of \( A \), denoted as \( \text{Adj}(A) \), is:

\[ \text{Adj}(A) = \begin{bmatrix} q & -n \\ -p & m \end{bmatrix}. \]

Given Condition:

We are given the condition:

\[ |A - d \cdot \text{Adj}(A)| = 0. \]

Substituting \( A \) and \( \text{Adj}(A) \) into the equation:

\[ A - d \cdot \text{Adj}(A) = \begin{bmatrix} m & n \\ p & q \end{bmatrix} - d \begin{bmatrix} q & -n \\ -p & m \end{bmatrix}. \]

This simplifies to:

\[ A - d \cdot \text{Adj}(A) = \begin{bmatrix} m - dq & n + dn \\ p + dp & q - dm \end{bmatrix}. \]

Determinant of \( A - d \cdot \text{Adj}(A) \):

The determinant is:

\[ |A - d \cdot \text{Adj}(A)| = \begin{vmatrix} m - dq & n + dn \\ p + dp & q - dm \end{vmatrix}. \]

Expanding the determinant:

\[ |A - d \cdot \text{Adj}(A)| = (m - dq)(q - dm) - (n + dn)(p + dp). \]

Simplify the terms:

\[ |A - d \cdot \text{Adj}(A)| = m \cdot q - m \cdot dm - dq \cdot q + d^2 \cdot q \cdot m - (n \cdot p + n \cdot dp + dn \cdot p + d^2 \cdot n \cdot p). \]

Using \( d = m \cdot q - n \cdot p \), the terms simplify further to:

\[ (1 + d)^2 = (m + q)^2. \]

Conclusion:

The correct answer is:

\[ (1 + d)^2 = (m + q)^2. \]