Let A be a 3 × 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained b

Q: Wait, why is $\det(2A) = 8 \det(A)$? Shouldn't it just be $2 \det(A)$?

That is a classic trap! Remember that the determinant of a $3 \times 3$ matrix represents the volume of a parallelepiped formed by its row vectors. If you multiply the matrix by 2, you are doubling the length of every vector in all three dimensions. Therefore, the volume scales by $2 \times 2 \times 2 = 2^3 = 8$. Always remember the property $\det(kA) = k^n \det(A)$ for an $n \times n$ matrix.

Q: Does adding $5R_3$ to $R_2$ change the determinant? It feels like it should!

It is counter-intuitive, but it does not! Geometrically, adding a multiple of one row to another is a shear transformation. Shear transformations preserve the volume of the parallelepiped. Algebraically, the elementary row operation $R_i \to R_i + cR_j$ is determinant-invariant. So, you can perform this operation as many times as you like without changing the determinant.

Q: Can I just assume $A$ is the identity matrix and solve it?

That is a brilliant 'JEE hack'! If the property holds for all matrices, it must hold for the simplest one. You could assume $A = \text{diag}(4, 1, 1)$ or similar, but understanding the properties is the 'true' path that will save you when the problem gets more abstract. Use the hack to verify, but master the properties to conquer!

Question

Let $A$ be a $3 	imes 3$ matrix with $\det(A) = 4$. Let $R_i$ denote the $i^{th}$ row of $A$. If a matrix $B$ is obtained by performing the operation $R_2 	o 2R_2 + 5R_3$ on $2A$, then $\det(B)$ is equal to:

Options:

(A) 64

(B) 16

(C) 80

(D) 128

Sigma Percentile · Accepted Answer

### Analyzing the Geometric Setup

Imagine you are standing in a 3D space, looking at a parallelepiped defined by three vectors, which are the rows of your matrix $A$. The determinant, $\det(A) = 4$, represents the volume of this shape.

We are not just dealing with numbers; we are dealing with the geometry of space. Understanding these transformations requires us to visualize how the volume of this parallelepiped changes under specific operations.

### The Scalar Trap

The first step is to transform $A$ into $2A$. Many students instinctively think that if you multiply the matrix by 2, the determinant doubles.

However, if you scale every dimension of a 3D object by 2, the volume increases by a factor of $2 	imes 2 	imes 2 = 8$. This follows the property:

$$\det(kA) = k^n \det(A)$$

With $n = 3$ and $k = 2$, we calculate:

$$\det(2A) = 2^3 	imes \det(A) = 8 	imes 4 = 32$$

We have successfully navigated the first trap.

### The Anatomy of Row Operations

Now, we perform the operation $R_2 	o 2R_2 + 5R_3$ on our new matrix $2A$. We can break this down into two simple, elegant steps.

First, we scale the second row by 2. Scaling a single row of a matrix by a constant $k$ scales the entire determinant by that same constant $k$.

Our determinant, which was 32, now becomes:

$$32 	imes 2 = 64$$

Second, we add 5 times the third row to the second row. Adding a multiple of one row to another is a shear transformation.

It changes the shape of our parallelepiped, but it leaves its volume completely unchanged. The determinant remains invariant under this operation.

### Final Synthesis

We have arrived at the end of our journey. We started with $\det(A) = 4$, scaled the entire matrix to get $\det(2A) = 32$, and then performed a row operation that effectively doubled the determinant while adding a shear that cost us nothing in terms of volume.

The final result is:

$$\det(B) = 64$$

It is not just about the calculation; it is about understanding that each row operation is a geometric transformation. You have mastered the logic, and that is the true essence of JEE Advanced mathematics.

Animated Solution for Mathematics - Matrices and Determinants: Let $A$ be a $3 \times 3$ matrix with $det (A) = 4$ . Let $R_{i}$ denote the $i^{t h}$ row of $A$ . If a matrix $B$ is obtained by performing the operation $R_{2} \to 2 R_{2} + 5 R_{3}$ on $2 A$ , then $det (B)$ is equal to:

Select Answer:

Visualized Solution

Visualizing the Given Matrix $A$

Forming Matrix $2 A$

Determinant Property: $det (k A)$

Calculating $det (2 A)$

Applying the Row Operation

Forming Matrix $B$

Effect of Scaling a Row

Calculating Intermediate Determinant

Effect of Adding a Row Multiple

The Sigma Insight: Properties of Determinants

Analyzing the Geometric Setup

The Scalar Trap

The Anatomy of Row Operations

Final Synthesis

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Visualized Solution

Visualizing the Given Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Forming Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2A

Determinant Property: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } det(kA)

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } det(2A)