Let $I$ be the identity matrix of order $3\times3$ and let $$A=\begin{pmatrix} \lambda & 2 & 3\\4 & 5 & 6\\7 & -1 & 2\end{pmatrix}$$ with $|A|=-1$. Let $B$ be the inverse of the matrix $\mathrm{adj}(A)\,\mathrm{adj}(A^2)$. Then $|\lambda B + I|$ is equal to:

Step 1: Simplify the expression for B

We are given that $B$ is the inverse of the matrix $\mathrm{adj}(A)\,\mathrm{adj}(A^2)$. We will use properties of adjoints and inverses to simplify this expression.

Step 2: Apply properties of adjoints and inverses

We use the property that $\mathrm{adj}(A^n) = (\mathrm{adj}(A))^n$. Also, recall that $\mathrm{adj}(A) = |A|A^{-1}$ and the inverse of a product of matrices is the product of their inverses in reverse order.

Step 3: Substitute and simplify B

First, we combine the adjoint terms: $\mathrm{adj}(A)\,(\mathrm{adj}(A))^2 = (\mathrm{adj}(A))^3$. Then, $B = ((\mathrm{adj}(A))^3)^{-1} = (\mathrm{adj}(A))^{-3}$. Since $|A|=-1$, we have $\mathrm{adj}(A) = |A|A^{-1} = -A^{-1}$. Substituting this, $B = (-A^{-1})^{-3} = (-1)^{-3}(A^{-1})^{-3} = -A^3$.

Step 4: Calculate $|\lambda B + I|$

Now we substitute $B = -A^3$ into the expression $|\lambda B + I|$.

Step 5: Use the property $|kA| = k^n|A|$

We know that $|A| = -1$. For a $3\times3$ matrix $M$ and a scalar $k$, the determinant of $kM$ is $k^3|M|$.

Step 6: Final calculation

The expression $|-\lambda A^3 + I|$ cannot be simplified further without knowing the value of $\lambda$ or the matrix $A$. The problem statement implies that the answer should be a specific numerical value, which suggests there might be a misunderstanding or missing information in the problem statement as presented. However, if we assume that the question implies a property that makes this expression simplify to a constant, it's not immediately obvious. Given the options usually provided in such problems, it's possible that the question expects a specific property of determinants that is not directly applicable here, or there's a typo in the question. Without further information or context, we cannot simplify $|-\lambda A^3 + I|$ to a numerical value.

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