Let A = [[4, -2], [α, β]]. If A² + γA + 18I = O, then det(A) is equal to (A) -18 (B) 18 (C) -50 (D)

Q: Why can't we just solve for $\alpha$ and $\beta$ by substituting the matrix $A$ into the equation?

You absolutely could, but that would be a long, tedious path! In JEE, time is your most precious resource. The Cayley-Hamilton theorem allows us to bypass the algebra of individual elements and look at the 'soul' of the matrix—its trace and determinant.

Q: Is the Cayley-Hamilton theorem valid for all square matrices?

Yes, it is! Whether it's a $2 \times 2$, $3 \times 3$, or an $n \times n$ matrix, the theorem holds true. It's a fundamental property of linear algebra that connects the matrix to its characteristic polynomial.

Q: Why do we write $\det(A)I$ instead of just $\det(A)$ in the matrix equation?

Great question! Remember, we are dealing with a matrix equation. $A^2$ is a matrix, $\gamma A$ is a matrix, and $O$ is a null matrix. To add $\det(A)$ (which is a scalar) to these, we must multiply it by the identity matrix $I$ to make it a matrix of the same dimensions. Dimensional consistency is key!

Question

Let $A = \begin{bmatrix} 4 & -2 \ \alpha & \beta \end{bmatrix}$. If $A^2 + \gamma A + 18I = O$, then $\det(A)$ is equal to

Options:

(A) -18

(B) 18

(C) -50

(D) 50

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a complex matrix equation. You see $A = \begin{bmatrix} 4 & -2 \ \alpha & \beta \end{bmatrix}$ and the daunting equation $A^2 + \gamma A + 18I = O$.

Your first instinct might be to start calculating powers of $A$, finding $\alpha$ and $\beta$, and diving into a sea of algebraic variables. But wait—stop.

In the world of JEE Advanced, the most elegant path is rarely the one that requires the most brute force. There is a hidden structure here, a mathematical heartbeat that we can tap into.

### The Cayley-Hamilton Revelation

Whenever you encounter an equation involving powers of a square matrix, your mind should immediately leap to the **Cayley-Hamilton Theorem**. This is one of the most powerful tools in your linear algebra toolkit.

It states that every square matrix satisfies its own characteristic equation. Think of it as the matrix's personal identity—a unique polynomial that the matrix itself must obey.

For a $2 	imes 2$ matrix, this characteristic equation is beautifully simple:

$$ \lambda^2 - \mathrm{Tr}(A)\lambda + \det(A) = 0 $$

where $\mathrm{Tr}(A)$ is the trace (the sum of the diagonal elements) and $\det(A)$ is the determinant.

### The Elegant Comparison

Now, let's apply this magic. According to the theorem, if we replace the scalar $\lambda$ with the matrix $A$, the equation becomes:

$$ A^2 - \mathrm{Tr}(A)A + \det(A)I = O $$

Notice how we attach the identity matrix $I$ to the constant term $\det(A)$? This is crucial for dimensional consistency—we are working in the realm of matrices, so every term must be a matrix!

Now, look at the equation provided in the problem: $A^2 + \gamma A + 18I = O$. We have two equations that both describe the behavior of matrix $A$.

By comparing them, we can see that the coefficients must match. The constant term in our derived equation is $\det(A)I$, and the constant term in the given equation is $18I$.

### The Final Triumph

Are you seeing the beauty of it? By simply equating the coefficients of $I$, we find that **$\det(A) = 18$**.

We didn't need to calculate $\alpha$, $\beta$, or $\gamma$. We didn't need to perform a single complex matrix multiplication.

We simply used the deep, geometric reality of the matrix to bypass the noise. This is the essence of JEE Advanced—not just solving, but understanding the underlying structure.

You have successfully navigated the trap and found the answer with elegance and precision. Keep this mindset, and no matrix equation will ever intimidate you again.

Animated Solution for Mathematics - Matrices and Determinants: Let $A = [4 α - 2 β]$ . If $A^{2} + γ A + 18 I = O$ , then $det (A)$ is equal to

Select Answer:

Visualized Solution

Given Matrix and Equation

Cayley-Hamilton Theorem

Characteristic Equation

Applying the Theorem

Comparing Equations

Focusing on the Constant Term

Final Result

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The Cayley-Hamilton Revelation

The Elegant Comparison

The Final Triumph

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=[4α​−2β​]. If A2+γA+18I=O, then det(A) is equal to