Problem 23 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant \(\Delta=\left|\begin{array}{ccc}2 a_{1} b_{1} & a_{1} b_{2}+a_{2} b_{1} & a_{1} b_{3}+a_{3} b_{1} \\ a_{1} b_{2}+a_{2} b_{1} & 2 a_{2} b_{2} & a_{2} b_{3}+a_{3} b_{2} \\ a_{1} b_{3}+a_{3} b_{1} & a_{3} b_{2}+a_{2} b_{3} & 2 a_{3} b_{3}\end{array}\right|\) is (A) 1 (B) \(-1\) (C) 0 (D) \(a_{1} a_{2} a_{3} b_{1} b_{2} b_{3}\)

Step-by-Step Solution

Answer

The determinant is 0.

Step 1: Recognize the Matrix

We have a 3x3 symmetric matrix. Symmetric matrices have the property that the terms on the left of the diagonal mirror those on the right.

Step 2: Simplify the Matrix Using Properties

Notice that the matrix can be seen as two vectors **a** = \((a_1, a_2, a_3)\) and **b** = \((b_1, b_2, b_3)\). The matrix is structured like an outer product: \[\begin{bmatrix}2a_1b_1 & a_1b_2 + a_2b_1 & a_1b_3 + a_3b_1 \a_1b_2 + a_2b_1 & 2a_2b_2 & a_2b_3 + a_3b_2 \a_1b_3 + a_3b_1 & a_2b_3 + a_3b_2 & 2a_3b_3\end{bmatrix}\] This hints at properties of products and determinants.

Step 3: Recognize Matrix Rank

The given matrix can be reduced to a case involving repeated outer products, suggesting it has a rank of 1 or 2 at most. For rank 1 matrices, determinants are 0.

Step 4: Evaluate Det using Properties

For any 3x3 symmetric matrix derived from such forms, provided the matrix can be written to resemble a vector product, it simplifies and leads to a determinant of 0, confirming our analysis.

Step 5: Determine the Answer

Given all previous steps, the determinant simplifies to 0. Therefore, the correct answer is option (C) 0.

Key Concepts

Matrix rank

The rank of a matrix is a fundamental concept in linear algebra. It refers to the dimension of the vector space generated by the rows or columns of the matrix. In simple terms, it gives us the number of linearly independent rows or columns.

• The rank helps us to understand the solutions of a system of linear equations. For instance, if the rank of a matrix equals the number of columns, the matrix is said to be full rank.
• A matrix with fewer independent columns or rows has a rank less than the full rank.

In our exercise, the matrix derived from the outer product suggests a rank of at most 1 or 2. This means the rows or columns can be expressed as combinations of fewer vectors. For matrices of rank 1, the determinant is naturally zero. This is crucial here as it helps us determine the solution is indeed option (C) 0.

Symmetric matrix

Symmetric matrices are special types of matrices where the elements are mirrored across the main diagonal. This means that the element at position \(i,j\) is equal to the element at position \(j,i\).

• They are quite common in mathematics and have wonderful properties, like always having real eigenvalues.
• The structure simplifies certain calculations, and their behavior is predictable when solving for determinants.

The matrix in our exercise is symmetric, which means the operation and understanding became easier. Symmetry plays a key role in determining properties like rank and determinant. Thus, recognizing symmetry helps us know that transformations will often maintain certain properties, allowing conclusions about the determinant without detailed computation.

Outer product

The outer product is a simple way to generate matrices using vector multiplication. When you take two vectors, say **a** and **b**, their outer product forms a matrix. The resulting matrix's dimensions equal the length of the first vector by the length of the second vector.

• The elements of this matrix are computed by multiplying each element of **a** with each element of **b**.
• This concept can be easier to visualize with actual numbers, but fundamentally provides a method to generate structured matrices.

In this exercise, the given matrix resembles the outer product of vectors **a** and **b** with additional coherent symmetries. This structured form hints at the rank and determinant properties. The matrix can be reshaped to expose its true form, showing why it results in a determinant of 0, as seen in rank 1 matrices constructed this way.