Problem 206 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the matrix \(\left[\begin{array}{lll}0 & a & 5 \\ 3 & 0 & b \\ c & 2 & 0\end{array}\right]\) is skew-symmetric, then find \(a, b, c\).

Step-by-Step Solution

Answer

The values of the coefficients are \(a = -3, b = -2, c = -5\)

Step 1: Write down the given matrix

Write down the given matrix \(A = \left[\begin{array}{lll}0 & a & 5 \ 3 & 0 & b \ c & 2 & 0\end{array}\right]\)

Step 2: Transpose the matrix

Transposing the matrix A, we get \(A^T = \left[\begin{array}{lll}0 & 3 & c \ a & 0 & 2 \ 5 & b & 0\end{array}\right]\)

Step 3: Formulate equations

As A is a skew-symmetric matrix, \(A^T\) should be equal to \(-A\). This gives us the following equations by comparing the respective elements: \(3 = -a\),\(c = -5\), and \(2 = -b\)

Step 4: Solve for a, b, c

Solving these equations we find that \(a = -3, b = -2, c = -5\)

Key Concepts

Matrix Transpose

A matrix transpose is a fundamental concept in linear algebra. It involves flipping a matrix over its diagonal, effectively switching the matrix's rows and columns.
When you transpose a matrix, you denote it with a superscript 'T', such as turning matrix A into \( A^T \).
In the context of a 3x3 matrix:

• The element in the first row, second column of the original matrix becomes the element in the second row, first column in the transposed matrix.
• For example, given the original element \( a_{12} \) in matrix A, \( a_{21} \) would be its counterpart in \( A^T \).

Understanding transposition is crucial for many purposes, such as in solving matrix equations or finding inverse matrices.
In our example, transposing is used to verify skew-symmetry, revealing essential relationships among the elements of the matrix.

IIT JEE

The IIT JEE (Indian Institutes of Technology Joint Entrance Examination) is a highly competitive exam. It is an entryway for engineering aspirants to secure seats in prestigious Indian Institutes of Technology.
The exam assesses a wide range of topics, including mathematics, physics, and chemistry.

• Mathematics plays a significant role, often involving concepts like matrices, calculus, and algebra.
• Problems related to matrices in the IIT JEE might involve properties like determinants, inverses, or symmetry.

Understanding concepts like skew-symmetric matrices can be crucial for this exam. They help in solving complex problems effectively and efficiently.
The exercise we're discussing here involves identifying and manipulating matrix properties, a skill frequently tested in these exams.

Equation Solving

Equation solving is a key skill in mathematics, required across various topics and exams.
In the context of matrices, it often includes setting up and solving equations derived from specific matrix properties.

• For a skew-symmetric matrix, each element of the transpose is the negative of the corresponding element in the original matrix, i.e., \( A^T = -A \).
• This property allows us to form equations by comparing elements from \( A \) and \( -A \).
• In our exercise, the equations were set up as \( 3 = -a, c = -5, \) and \( 2 = -b \).

Solving such equations usually involves basic algebra.
Once the equations are established, they are solved to find unknown elements like \( a, b, \) and \( c \).
Practicing these skills helps students tackle more complex mathematical problems effortlessly.