Q.If the value of a third order determinant is $12$, then the value of the determinant formed by replacing each element by its co-factor will be $144$.

Concept: Determinant Evaluation Using Identities — specifically, the relation between a determinant and its cofactor matrix.

For a square matrix $A$ of order $n$, if $C$ is the cofactor matrix, then $|C| = |A|^{n-1}$. Here $n = 3$, so $|C| = |A|^{2}$.

Step 1: The given determinant is $|A| = 12$. …

The problem asks: if a third-order determinant has value $12$, what is the value of the determinant formed by replacing each element by its cofactor? This is a classic result in determinant theory, and the answer follows directly from a fundamental property relating a matrix and its adjoint.

Let’s understand why this works. For any square matrix $A$ of order $n$, the matrix of cofactors (often denoted $C$) has a determinant that is related to $\det(A)$ by a simple power law. Specifically, if $A$ is $n \times n$, then $\det(\text{cofactor matrix of } A) = (\det A)^{n-1}$. For $n=3$, this becomes $(\det A)^{2}$.

Why? Because the cofactor matrix is intimately linked to the adjoint (adjugate) of $A$. The adjoint of $A$, written $\text{adj}(A)$, is the transpose of the cofactor matrix. A well-known identity is:

Taking determinants on both sides gives:

Since $\det(\text{adj}(A)) = \det(\text{cofactor matrix})$ (transpose doesn’t change determinant), we get:

Now apply this to the given problem.

1. Identify the order: The determinant is third order, so $n = 3$. The original determinant value is $\Delta = 12$. …

Method: Finding the Determinant of the Cofactor (or Adjoint) Matrix

Use this method whenever a question gives $|A|$ for an $n\times n$ matrix and asks for the determinant of the matrix formed by its cofactors (or its adjoint).

Steps

Step 1: Recall the defining identity linking $A$ and $\operatorname{adj}(A)$

where $\operatorname{adj}(A)$ is the transpose of the cofactor matrix.

Step 2: Take determinants of both sides

using $|kI_n| = k^n$ for a scalar matrix of order $n$.

Step 3: Solve for $|\operatorname{adj}(A)|$

Since $A$ is non-singular ($|A|\ne0$), divide both sides by $|A|$:

Step 4: Relate the adjoint's determinant to the cofactor matrix's determinant …

Common Mistakes

Mistake 1: Assuming the cofactor-matrix determinant equals $|A|$ itself

Why it's wrong: it's tempting to think replacing entries with cofactors "shouldn't change" the determinant, but the correct relationship is a POWER law, $|A|^{n-1}$, not equality — for $n>2$ these give very different numbers. Correct approach: always derive from $A\cdot\operatorname{adj}(A)=|A|I_n$ rather than assuming equality.

Mistake 2: Using the wrong exponent (forgetting it's $n-1$, not $n$) …