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

To find the relation between the determinant d and the elements of the matrix A, we evaluate the given determinant condition:

$$\left| A - d(\text{Adj } A) \right| = 0$$

The matrix A and its determinant d are defined as:

$$A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}, \quad d = |A| = mq - np$$

The adjoint of a 2x2 matrix is obtained by swapping the diagonal elements and changing the signs of the off-diagonal elements:

$$\text{Adj } A = \begin{pmatrix} q & -n \\ -p & m \end{pmatrix}$$

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Step 1: Express the matrix $$A - d(\text{Adj } A)$$

Multiplying the adjoint matrix by d:

$$d(\text{Adj } A) = \begin{pmatrix} dq & -dn \\ -dp & dm \end{pmatrix}$$

Now, subtracting this from matrix A:

$$A - d(\text{Adj } A) = \begin{pmatrix} m & n \\ p & q \end{pmatrix} - \begin{pmatrix} dq & -dn \\ -dp & dm \end{pmatrix} = \begin{pmatrix} m - dq & n(1 + d) \\ p(1 + d) & q - dm \end{pmatrix}$$

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Step 2: Evaluate the determinant and set it to zero

$$\left| A - d(\text{Adj } A) \right| = \begin{vmatrix} m - dq & n(1 + d) \\ p(1 + d) & q - dm \end{vmatrix} = 0$$

Expanding the determinant:

$$(m - dq)(q - dm) - np(1 + d)^2 = 0$$

$$mq - m^2d - q^2d + mqd^2 - np(1 + d)^2 = 0$$

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Step 3: Simplify using the definition of d

Grouping the terms with mq:

$$mq(1 + d^2) - d(m^2 + q^2) - np(1 + d)^2 = 0$$

We know that np = mq - d. Substituting this into the equation:

$$mq(1 + d^2) - d(m^2 + q^2) - (mq - d)(1 + 2d + d^2) = 0$$

$$mq + mqd^2 - dm^2 - dq^2 - (mq + 2mqd + mqd^2 - d - 2d^2 - d^3) = 0$$

Canceling out the common terms $$mq$$ and $$mqd^2$$:

$$-dm^2 - dq^2 - 2mqd + d + 2d^2 + d^3 = 0$$

Dividing the entire equation by $$-d$$ (since $$d \neq 0$$):

$$m^2 + q^2 + 2mq - 1 - 2d - d^2 = 0$$

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Step 4: Form the perfect squares

Recognizing the algebraic expansions for $$(m + q)^2$$ and $$(1 + d)^2$$:

$$(m + q)^2 - (1 + 2d + d^2) = 0$$

$$(m + q)^2 - (1 + d)^2 = 0$$

$$(1 + d)^2 = (m + q)^2$$

Therefore, the final relationship is equal to $$(1 + d)^2 = (m + q)^2$$.