Let A be a 2 x 2 matrix with non-zero entries and let A² = I, where I is 2 x 2 identity matrix. Defi

Q: Wait, why do we reject the negative root here? Or why can't we just use energy conservation?

This is a great question! In this specific problem, we are not dealing with energy conservation, but rather the algebraic properties of involutory matrices (matrices where $A^2 = I$). We don't 'reject' a root; rather, we derive the trace and determinant directly from the matrix multiplication equations. The constraint that all entries are non-zero is the key that forces the trace to be zero.

Q: Why is the determinant -1 and not 1? I thought $|A^2| = |I| = 1$, so $|A|$ must be 1.

You are absolutely right that $|A|^2 = 1$, which means $|A|$ could be $1$ or $-1$. However, the specific constraint that all entries are non-zero forces the matrix to have a specific form. By substituting $d = -a$ and $bc = 1 - a^2$ into the determinant formula $|A| = ad - bc$, we find that the $a^2$ terms cancel out, leaving us with exactly $-1$. It is a beautiful, rigorous result!

Q: Do I really need to write out the full matrix multiplication? Can't I just use properties of the trace?

While properties of the trace are powerful, writing out the multiplication is the safest way to ensure you don't miss the off-diagonal constraints. In JEE Advanced, the 'trap' is often in the details—like the off-diagonal elements being zero. By writing it out, you see the $b(a+d) = 0$ condition clearly, which is the heart of the problem.

Question

Let $A$ be a $2 	imes 2$ matrix with non-zero entries and let $A^2 = I$, where $I$ is $2 	imes 2$ identity matrix. Define $Tr(A) = $ sum of diagonal elements of $A$ and $|A| = $ determinant of matrix $A$. \ Statement-1: $Tr(A) = 0$. \ Statement-2: $|A| = 1$.

Options:

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B) Statement-1 is true, Statement-2 is false.

(C) Statement-1 is false, Statement-2 is true.

(D) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are looking at a $2 	imes 2$ grid of numbers, a matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$. The problem states that $A^2 = I$, where $I$ is the identity matrix.

These are known as **involutory matrices**, which act like reflections in the realm of linear algebra. We are given the critical constraint that all entries $a, b, c, d$ are non-zero.

### The Matrix Multiplication Journey

To understand the condition $A^2 = I$, we perform the matrix multiplication:

$$A^2 = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} a^2+bc & b(a+d) \ c(a+d) & bc+d^2 \end{bmatrix}$$

We set this result equal to the identity matrix $\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$.

Focusing on the off-diagonal elements, the top-right element is $b(a+d)$. Since this must equal $0$ and we are given $b 
eq 0$, the only logical conclusion is that $a+d = 0$.

Because the trace of the matrix is defined as $Tr(A) = a+d$, we have proven that $Tr(A) = 0$. Therefore, **Statement-1 is true**.

### The Determinant Trap

Now, we evaluate Statement-2 by calculating the determinant $|A| = ad - bc$. From our multiplication result, we know that $a^2 + bc = 1$, which implies $bc = 1 - a^2$.

We also established that $a+d = 0$, which means $d = -a$. Substituting these values into the determinant formula yields:

$$|A| = a(-a) - (1 - a^2)$$

Expanding this expression, we get:

$$|A| = -a^2 - 1 + a^2$$

The $a^2$ terms cancel out with precision, leaving us with $|A| = -1$.

Since Statement-2 claims $|A| = 1$, it is **false**. This serves as a reminder to rely on rigorous derivation rather than intuition when navigating JEE-style problems.

Select Answer:

Visualized Solution

Defining Matrix $A$

Setting up $A^{2}$

Computing $A^{2}$

Equating to Identity Matrix $I$

Extracting Off-Diagonal Equations

Applying the Non-Zero Condition

Evaluating Statement-1

Extracting Diagonal Equations

Setting up the Determinant $∣ A ∣$

Substituting Known Values

Calculating the Determinant

Evaluating Statement-2

Final Conclusion

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The Matrix Multiplication Journey

The Determinant Trap

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