Q.If 𝐴 is a square matrix of order 4 and |𝑎𝑑𝑗 𝐴| = 27, then 𝐴 (𝑎𝑑𝑗 𝐴) is equal to
(A) 3
(B) 9
(C) 3 𝐼
(D) 9 𝐼

Concept: Adjoint Matrix Property — For any square matrix $A$ of order $n$,

$A (\text{adj } A) = |A| I_n$, and $|\text{adj } A| = |A|^{n-1}$.

Step 1: Given $n = 4$ and $|\text{adj } A| = 27$.

Using $|\text{adj } A| = |A|^{n-1}$, we have $|A|^{3} = 27$. …

We start with a fundamental property of adjoint matrices: for any square matrix $A$ of order $n$, the product $A (\text{adj } A)$ equals $|A| I$, where $I$ is the identity matrix of the same order. This is not a trick — it's the defining relationship that makes the adjoint useful for finding inverses. So the question reduces to: what is $|A|$?

We are told $|\text{adj } A| = 27$ and $A$ is of order 4. There is a well-known formula connecting the determinant of the adjoint to the determinant of the original matrix: $|\text{adj } A| = |A|^{n-1}$, where $n$ is the order. For $n = 4$, this becomes $|\text{adj } A| = |A|^3$.

1. Apply the adjoint determinant formula. Since $|\text{adj } A| = |A|^{4-1} = |A|^3$, and we know $|\text{adj } A| = 27$, we have:

Taking the real cube root (determinants are real numbers here), we get:

2. Use the fundamental product property. Now, $A (\text{adj } A) = |A| I$. Substituting $|A| = 3$ and noting $I$ is the $4 \times 4$ identity matrix:

3. Interpret the result. The expression $3I$ is a scalar multiple of the identity matrix — not a scalar number. Among the options, (A) 3 and (B) 9 are scalars, not matrices. Option (D) is $9I$, which would require $|A| = 9$. Only option (C) $3I$ matches. …

Method: Order-Relation Route from |adj A| to A(adj A)

This method solves any problem where you're given information about |adj A| (or vice versa) for a square matrix of known order, and asked for A(adj A), |A|, or a related quantity — without ever knowing the entries of A.

Steps

Step 1: Write down the two governing adjoint identities

For any square matrix $A$ of order $n$, two identities connect $A$, $\operatorname{adj}(A)$, and their determinants:

The first is a matrix identity; the second is a scalar identity that follows from taking determinants of the first. Recognise which one gives you a path from the known quantity to the unknown one.

Step 2: Use the order-relation formula to isolate |A| …

Common Mistakes

Mistake 1: Using the wrong exponent in $|\text{adj }A|=|A|^{n-1}$

Why it's wrong: students often write $|\text{adj }A|=|A|^n$ (matching the order of the matrix instead of order minus one), which for $n=4$ gives $|A|^4=27$ — an equation with no clean real solution. Correct approach: always use exponent $n-1$; here $n=4$ gives $|A|^3=27$, so $|A|=3$.

Mistake 2: Treating $A(\text{adj }A)$ as if the answer must be a plain number

Why it's wrong: $A(\text{adj }A)$ is always a matrix (equal to $|A|I_n$), never a bare scalar — so options like "3" or "9" without the identity matrix can never be correct once the matrix has order greater than $1$. Correct approach: always keep the $I_n$ factor; the answer here is the matrix $3I$, not the number $3$. …