Problem 55 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the determinant \(\left|\begin{array}{cccc}0 & x & y & z \\ -x & 0 & c & b \\ -y & -c & 0 & a \\ -z & -b & -a & 0\end{array}\right|\) is (A) \((a x+b y+c z)^{2}\) (B) \((a x-b y+c z)^{2}\) (C) \((a x+b y-c z)^{2}\) (D) None of these

Step-by-Step Solution

Answer

The determinant is \((ax + by + cz)^2\), which corresponds to option (A).

Step 1: Recognize the Matrix is Skew-Symmetric

The given matrix is a 4x4 skew-symmetric matrix. A matrix is skew-symmetric if \( A_{ij} = -A_{ji} \) for all \( i, j \). In this matrix, each element \( A_{ij} \) equals \( -A_{ji} \), implying it is indeed skew-symmetric.

Step 2: Calculate Determinant of Skew-Symmetric Matrix

The determinant of a skew-symmetric matrix of odd order is zero, and for even order, it can be a perfect square. In this case, the matrix is 4x4 (even order), and based on properties of such matrices, its determinant can be a square of a polynomial expression involving its elements.

Step 3: Determine Polynomial Expression

By noticing the pattern of the elements and past experience with similar matrices, we determine that the expression for the determinant is based on linear combinations of the last column with the first row terms that are cyclic, i.e., the expression \((ax + by + cz)\).

Step 4: Conclude the Determinant Value

Thus, the determinant of this skew-symmetric matrix is \((ax + by + cz)^2\), based on previously known results about determinant evaluations of skew-symmetric matrices of order 4.

Key Concepts

Determinant Properties

The determinant is a numerical value derived from a square matrix that gives insights into various matrix properties. One crucial property of determinants is their behavior towards skew-symmetric matrices. A skew-symmetric matrix satisfies the condition \( A_{ij} = -A_{ji} \) for all \( i \) and \( j \), meaning that its elements across the diagonal are negatives of each other. An important aspect of skew-symmetric matrices is how their determinant behaves based on the matrix order:

• For odd order skew-symmetric matrices, the determinant is always zero.
• For even order skew-symmetric matrices, the determinant is often a perfect square.

If we consider the given 4x4 matrix, which is of an even order, its determinant follows as a perfect square due to these properties. This can guide us towards finding or verifying solutions connected to these matrices.

Polynomial Expression

A polynomial expression in the context of matrix determinanta can significantly simplify calculations. For skew-symmetric matrices of even order, the determinant can be expressed in terms of polynomial expressions involving the elements of the matrix. In the given problem, the solution provides the polynomial expression \((ax + by + cz)\), which gives form to the determinant value.
To arrive at this expression, recognize the pattern between the matrix's rows and columns. The elements exhibit a type of cyclic symmetry that allows us to identify \((ax + by + cz)\) as a relevant linear combination of terms. This expression not only aligns with known results of skew-symmetric matrices but also makes computations more manageable.
Understanding how to derive such expressions is essential for efficiently solving determinant problems involving even order skew-symmetric matrices.

Matrix Order

Matrix order refers to the dimensions of a matrix, represented as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. In determinant problems, knowing the order of the matrix is fundamental. It influences properties such as invertibility and special characteristics of the determinant.
In our exercise, the matrix is of order 4x4, meaning it has four rows and four columns. With skew-symmetric matrices, this even order property is crucial as it indicates that the determinant may be a perfect square, unlike those odd order ones, which result in a determinant of zero.
Recognizing the matrix order helps us apply specific properties such as the skew-symmetric determinant property, making it easier to approach and solve the problem.