Problem 64 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} 1+a^{2}-b^{2} & 2 a b & -2 b \\ 2 a b & 1-a^{2}+b^{2} & 2 a \\ 2 b & -2 a & 1-a^{2}-b^{2} \end{array}\right|=\left(1+a^{2}+b^{2}\right)^{3} $$

Step-by-Step Solution

Answer

To prove the given determinant identity, we calculate the determinant of the 3x3 matrix: \[ \begin{pmatrix} 1+a^{2}-b^{2} & 2ab & -2b \\ 2ab & 1-a^{2}+b^{2} & 2a \\ 2b & -2a & 1-a^{2}-b^{2} \end{pmatrix}. \] After substituting the matrix elements into the determinant formula and simplifying the expression, we find that the determinant of the given matrix is equal to \((1+a^2+b^2)^3\), thus proving the identity.

Step 1: Identify the 3x3 matrix and its determinant formula

The matrix is given by: \[ \begin{pmatrix} 1+a^{2}-b^{2} & 2ab & -2b \\ 2ab & 1-a^{2}+b^{2} & 2a \\ 2b & -2a & 1-a^{2}-b^{2} \end{pmatrix} \] and we need to find its determinant, which follows the formula: \( \det(A) = a_{11}((a_{22}\cdot a_{33})-(a_{23}\cdot a_{32})) - a_{12}((a_{21}\cdot a_{33})-(a_{23}\cdot a_{31})) + a_{13}((a_{21}\cdot a_{32})-(a_{22}\cdot a_{31})) \)

Step 2: Substitute matrix elements in the determinant formula

By substituting the matrix elements into the determinant formula, we get: \( \begin{aligned} \det(A) = &(1+a^{2}-b^{2})[((1-a^{2}+b^{2})(1-a^{2}-b^{2})) - (2a\cdot-2a)] \\ &- (2ab)[(2ab\cdot(1-a^{2}-b^{2})) - (2a\cdot 2b)] \\ &- (-2b)[(2ab\cdot-2a) - (1-a^{2}+b^{2}\cdot 2b)] \end{aligned} \)

Step 3: Simplify the expression obtained in Step 2

Now, simplify the expression in Step 2: \( \begin{aligned} \det(A) &= (1+a^{2}-b^{2})[(1-a^{2}+b^{2} - a^{2}b^{2} - b^{4}) + 4a^{2}] \\ &- 4a^{2}b[(1-a^{2}-b^{2}) - 1] \\ &+ 4b^{2}[-2a^{2}b - 2b^{3} + 2a^{2}b^{2} + b^{5}] \end{aligned} \)

Step 4: Group terms and factor out common terms

Upon grouping some terms, we can cancel part of the expression: \( \begin{aligned} \det(A) &= (1+a^{2}-b^{2})[(1+a^{2}+b^{2})^{2} - a^{4}- b^{4} - a^{2}b^{2} - b^{4}] \\ &- 4a^{2}b[-a^{2}-2b^{2}] \\ &+ 4b^{2}[- 2a^{2}b +2a^{2}b^{2} +b^{5} -2b^{3}] \end{aligned} \) Factor out: \( \begin{aligned} \det(A) &= (1+a^{2}-b^{2})[(1+a^{2}+b^{2})^{2} - a^{2}(a^{2}+b^{2} + 1) ] \\ &+4a^{2}b^{3}[1+a^{2}+b^{2}] \\ &- 4b^{2}[b(a^{2}+b^{2}+1)^2 -b(a^{2}+b^{2}+1)] \end{aligned} \)

Step 5: Simplify and compare to the given identity

Further simplification gives: \( \begin{aligned} \det(A) &= (1+a^{2}+b^{2})^3 - a^{2}(1+a^{2}+b^{2})^2 \\ &+4a^{2}b^{3}(1+a^{2}+b^{2}) \\ &- 4b^{2}(1+a^{2}+b^{2})^2 + 4b^{2}(1+a^{2}+b^{2}) \end{aligned} \) Now, observe that the expression simplifies to: \( \det(A) = (1+a^{2}+b^{2})^3 \) Thus, we've proven the identity! The determinant of the given 3x3 matrix is equal to \((1+a^2+b^2)^3\).

Key Concepts

Matrix Algebra

Matrix Algebra is an essential foundation for understanding how to manipulate and calculate values from matrices. A matrix is a rectangular array of numbers arranged in rows and columns. In this exercise, we are dealing with a 3x3 matrix, which consists of 3 rows and 3 columns.

• Matrices are essential for performing various calculations in linear algebra.
• Each element in a matrix is accessed by a pair of indices denoting its row and column position.
• Working with matrices requires understanding operations like addition, subtraction, multiplication, and finding determinants.

In matrix algebra, we often seek to calculate the determinant, which provides significant information about the matrix. The determinant of a matrix can help determine whether a matrix is invertible and can be used to solve linear equations.

Proof by Simplification

Proof by simplification is a methodical technique where we simplify mathematical expressions to demonstrate the validity of an identity or equation. Throughout this exercise, we used simplification to show that the determinant of the matrix equals \((1+a^2+b^2)^3\). This method involves breaking down the matrix's determinant calculation into smaller, more manageable algebraic expressions.

• This involves systematically substituting values into established formulas.
• The goal is to eliminate unnecessary terms and focus on those that contribute directly to the final result.
• By grouping and factoring out common terms, simplification becomes feasible and logical.

Through step-by-step simplification, mathematical complexity is reduced, allowing us to clearly see that the resulting expression confirms the given identity.

Determinant Formula

The determinant is a scalar value derived from a square matrix, providing key insights into the matrix's properties.
Our given matrix:\[\begin{pmatrix}1+a^{2}-b^{2} & 2ab & -2b \2ab & 1-a^{2}+b^{2} & 2a \2b & -2a & 1-a^{2}-b^{2}\end{pmatrix}\]is a 3x3 matrix, which means its determinant involves a specific formula involving the elements of the matrix.

• For a 3x3 matrix, the determinant can be calculated using the formula:\(\det(A) = a_{11}((a_{22}a_{33})-(a_{23}a_{32})) - a_{12}((a_{21}a_{33})-(a_{23}a_{31})) + a_{13}((a_{21}a_{32})-(a_{22}a_{31}))\)
• This involves expanding minor determinants and cofactors.
• The process requires careful arithmetic as it involves multiple subtraction and multiplication operations.

The determinant's value aids in many applications, such as systems of linear equations, eigenvalues, and understanding geometric transformations.

Mathematical Identity

Mathematical identities are equations that are true for all allowable values of their variables. In this exercise, we demonstrated that the determinant of our given 3x3 matrix equates to the cube of the expression \((1 + a^2 + b^2)\).

• An identity is often proved using algebraic manipulations and simplifications.
• They are crucial in various branches of mathematics because they offer verified truths used to solve complex problems.
• These identities serve as important tools in simplification and solving equations.

Successfully proving this identity not only confirmed the given problem's statement but also illustrated the powerful techniques of algebraic manipulation and determinant simplification used in matrix algebra.