Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Suppose, \(a, b, c\) are real numbers such that \(a b c=1\). If the matrix \(A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]\) is such that \(A^{\prime} A=I\), then the value of \(a^{3}+b^{3}+c^{3}\) is (A) 1 (B) 2 (C) 3 (D) 4

Step-by-Step Solution

Answer

The value of \(a^3 + b^3 + c^3\) is 3.

Step 1: Understand the Given Matrix and Condition

We are given a matrix \(A\) and an equation \(A' A = I\) where \(A'\) is the transpose of \(A\). This indicates that \(A\) is an orthogonal matrix. For an orthogonal matrix \(A\), \(A^{-1} = A'\), and both \(A' A = I\) and \(A A' = I\) hold true.

Step 2: Express the Matrix and Its Transpose

The given matrix \(A\) is:\[A = \left[\begin{array}{ccc} a & b & c \ b & c & a \ c & a & b \end{array}\right]\]The transpose of \(A\) is:\[A' = \left[\begin{array}{ccc} a & b & c \ b & c & a \ c & a & b \end{array}\right]\]Since \(A\) and \(A'\) are identical, this is a symmetric matrix.

Step 3: Calculate \(A' A\)

The product \(A' A\) is computed as:\[A' A = \begin{bmatrix} a & b & c \ b & c & a \ c & a & b \end{bmatrix} \begin{bmatrix} a & b & c \ b & c & a \ c & a & b \end{bmatrix}\]Perform matrix multiplication and simplify.

Step 4: Perform Matrix Multiplication

Calculate each element of the resulting matrix:- First row, first column: \(a^2 + b^2 + c^2\)- First row, second column: \(ab + bc + ca\)- First row, third column: \(ac + ba + cb\)- Second row, and so forth, The product \(A' A\) must equal the identity matrix \(I\).

Step 5: Equate to Identity Matrix

The matrix equation \(A' A = I\) implies:\[\begin{bmatrix} a^2 + b^2 + c^2 & ab + bc + ca & ac + ba + cb \ ab + bc + ca & b^2 + c^2 + a^2 & ... \ ... & ... & ... \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]This implies that \(a^2 + b^2 + c^2 = 1\) and \(ab + bc + ca = 0\).

Step 6: Use Factorization Identity

Using the identity \(x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)\), substitute \(x=a\), \(y=b\), and \(z=c\).Since \(ab + bc + ca = 0\), the second factor becomes \((a+y+z)(x^2+y^2+z^2)\).

Step 7: Evaluate \(a^3+b^3+c^3\)

Since \(a b c = 1\), substituting these conditions gives:\[a^3 + b^3 + c^3 - 3(1) = 0\]Thus, \(a^3 + b^3 + c^3 = 3\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. When multiplying two matrices, particularly if both are square matrices like in our exercise, we proceed by taking the dot product of rows and columns. In the exercise, we find:

• Each entry of the resulting matrix from multiplying matrices \(A'\) and \(A\), such as \(a^2 + b^2 + c^2\), is calculated by summing the products of corresponding entries in the row of \(A'\) and a column of \(A\).
• Matrix multiplication requires the first matrix's number of columns to match the second matrix's rows, which is compatible in square matrices.
• The resulting product, when performed correctly, forms another matrix showcasing the importance of analyzing each entry individually.

Here, the calculation of \(A' A\) results in an identity matrix which indicates that \(A\) is an orthogonal matrix. This property plays a critical role in solving problems that deal with orthogonal transformations in linear spaces.

Matrix Transpose

The transpose of a matrix, denoted by \(A'\), involves flipping a matrix over its diagonal. This means switching the matrix's row and column indices for every element.

• In the exercise provided, notice how matrix \(A\) remains the same after transposing. This retention of structure is due to symmetry.
• The transpose operation effectively affects non-square matrices more, providing critical symmetry operations in square matrices like identity and orthogonal matrices.
• One key point to emphasize is that for a symmetric matrix, its transpose equals the original matrix: \(A = A'\). This is why database systems can sometimes store symmetric matrices more efficiently.

Recognizing these simple operations within matrices like transpose can significantly enhance the understanding of complex problems derived from their inverted operations.

Symmetric Matrix

A symmetric matrix has the property where \(A = A'\), meaning it is equal to its transpose.

• In practice, this symmetry signifies that the elements about the main diagonal mirror each other, simplifying many matrix operations.
• The given matrix \(A\) in the exercise is symmetric, evident from its identical rows and columns post-transposition.
• In linear algebra, symmetric matrices are crucial due to their simplified eigenvalue computation and signal processing applications.

Symmetry in matrices not only simplifies computations but also highlights theoretical importance. In this exercise, recognizing the matrix as symmetric allowed for straightforward problem solving, using properties of orthogonal matrices to determine the expression for \(a^3 + b^3 + c^3\). Understanding such relationships enhances both practical and theoretical aspects of matrix applications.