Problem 46 - Algebra for JEE ADVANCED

Question

Suppose \(D=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) and \(D^{\prime}=\left|\begin{array}{lll}a_{1}+p b_{1} & b_{1}+q c_{1} & c_{1}+r a_{1} \\ a_{2}+p b_{2} & b_{2}+q c_{2} & c_{2}+r a_{2} \\ a_{3}+p b_{3} & b_{3}+q c_{3} & c_{3}+r a_{3}\end{array}\right|\). Then a. \(D^{\prime}=D\) b. \(D^{\prime}=D(1-p q r)\) c. \(D^{\prime}=D(1+p+q+r)\) d. \(D^{\prime}=D(1+p q r)\)

Step-by-Step Solution

Answer

a. \(D^{\prime}=D\)

Step 1: Identify the transformation

Examine the given determinants. Recognize that the transformation from \(D\) to \(D^{\prime}\) involves adding multiples of rows to rows using constants \(p\), \(q\), and \(r\). This indicates alteration using elementary row operations.

Step 2: Apply properties of determinants

Remember that multiplying a row of a determinant by a constant \(k\) will multiply the determinant by \(k\). However, here a combination of rows is involved. Such operations (adding a multiple of one row to another) do not change the determinant's value.

Step 3: Analyze effects of operations

Since each element \(a_i + p b_i\), \(b_i + q c_i\), \(c_i + r a_i\) in \(D^{\prime}\) is linear combination of elements from \(D\), the operations performed have no effect on the determinant's value because they are linear additions without affecting determinant by scalars in the whole system.

Step 4: Conclude the determinant's equality

From the properties of determinants and the transformations described, conclude that \(D^{\prime} = D\). This is due to the invariance of determinant through linear transformations of this sort.

Key Concepts

Determinant Properties

Understanding the properties of determinants is key in many areas of algebra. A determinant is a scalar value that you can compute from the elements of a square matrix. It provides important insights into the matrix, such as whether the matrix is invertible.
Some fundamental properties of determinants that are particularly useful include:

• Linearity: The determinant is a linear function of each row when all other rows are kept constant. This means you can add multiples of rows together without changing the determinant value.
• Row switching property: If you swap two rows in a matrix, the determinant changes its sign.
• Scalar multiplication: Multiplying an entire row by a constant will multiply the determinant by that same constant.
• Zero property: If two rows are identical, the determinant is zero.
• Additivity: You can add rows in the matrix, and the determinant still keeps the same value, as long as it’s a simple transformation.

Applying these properties helps to figure out why certain operations on rows, such as adding a multiple of one row to another, don’t affect the determinant as seen in this exercise example.

Elementary Row Operations

Elementary row operations are tools used for manipulating matrices. They are useful in solving systems of linear equations, finding determinants, and inverting matrices. These operations include:

• Switching rows: Swapping any two rows. This operation changes the determinant's sign.
• Multiplying a row by a non-zero scalar: This scales the determinant by the same factor.
• Adding a multiple of one row to another: This does not affect the determinant’s value and is a crucial feature for maintaining the matrix during Gaussian elimination or similar processes.

In the exercise provided, adding multiples of rows is central. This operation preserves the determinant. By adding a linear combination of matrix rows to another row, the original determinant's value remains unaffected. Hence, using these operations without altering the scalar value is essential for matrix manipulation.

Linear Transformations in Matrices

Linear transformations are mappings between vector spaces that preserve vector addition and scalar multiplication. In matrix terms, they transform matrices while maintaining essential characteristics such as linearity and determinant properties.
For example:

• Matrix Multiplication: Transforming a matrix with another through multiplication while preserving its linear structure.
• Linear combinations: Summing or scaling the rows or columns of matrices, which characterize the transformations seen during operations without changing the determinant's value.

In the context of determinants, certain transformations, specifically linear combinations like those seen in the exercise from matrix D to D', involve adding multiples of another row to a row. This keeps the determinant unchanged. This invariance is crucial; it implies that such linear transformations do not influence the ultimate calculation of a matrix's determinant, and potentially its invertibility, rank, and other properties are preserved, providing solid fundamentals in matrix algebra.