Problem 45 - 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

The answer is a. \(D^{\prime}=D\), due to determinant properties unchanged by adding column multiples to other columns.

Step 1: Understand the Problem

We are given two determinants, \(D\) and \(D'\). We want to determine how \(D'\) is related to \(D\). The matrices in the determinants differ by the terms \(p b_i\), \(q c_i\), and \(r a_i\) added to elements in different positions. We must explore the effect of these additions on the determinant.

Step 2: Simplify \(D'\) Expansions

To see how the modifications affect the determinant, let's consider the properties of determinants. Adding a multiple of a column to another column does not change the determinant. Check if the additions to the terms effectively show this property.

Step 3: Apply the Property of Determinants

We know from determinant properties that adding \(p\) times a column to another column doesn't change a determinant's value. Applying similar reasoning, for each row transformation in \(D'\), these transformations should not modify the determinant's value if added appropriately across columns.

Step 4: Analyze Each Addition's Effect

For each modification \(a_i + p b_i\), \(b_i + q c_i\), \(c_i + r a_i\), inspect if adding \(pb_i\), \(qc_i\), and \(ra_i\) cancels each other's effect considering the determinant's linearity and cross-terms.

Step 5: Check Special Cases

Sometimes calculating for specific small matrices such as 2x2 and generalizing can help identify patterns that support or negate transformations. Check special cases for pure numerical answers with specific \(p, q, r\) values to look for evidence of determinant relations.

Step 6: Formulate Based on Properties

Using determinant properties such as linearity and the effects of adding proportional rows and columns:- Analyze results that form or negate patterns of \(1+pqr\), \(1+p+q+r\), etc., showing similar results in different setups.

Key Concepts

Properties of Determinants

Determinants have several important properties that can simplify calculations and help us understand matrix operations. One of the key properties is that the determinant of a triangular matrix (with either zeros above or below the main diagonal) is simply the product of its diagonal entries. This is particularly useful when we consider row-reduced matrices.
Another property is that swapping two rows (or columns) of a matrix changes the sign of its determinant. This can be crucial in understanding how different row and column configurations affect the overall value of the determinant. Furthermore, if all elements of a row or column are multiplied by a constant, the determinant is also multiplied by that constant. Hence, if the whole matrix is scaled by a scalar, the determinant is exponentially affected (as each row's scaling builds upon the previous).
A significant attribute of determinants is that if two rows or columns are identical, the determinant is zero. This tells us that such a matrix is singular and does not have an inverse. These properties culminate in the ability to judge the invertibility and consistency of systems represented by matrices using determinants.

Effects of Row and Column Operations

Matrix operations on rows and columns can significantly impact a determinant, though some effects are more apparent than others. Elementary row operations include swapping rows, multiplying a row by a scalar, and adding a multiple of one row or column to another. These operations maintain or transform the determinant in specific predictable ways.

• Swapping two rows or columns changes the sign of the determinant.


• Multiplying a row by a scalar also multiplies the determinant by the same scalar.


• Adding a multiple of one row to another does not alter the determinant.

These rules are incredibly helpful for analyzing and simplifying the calculation of determinants, especially for large matrices. Understanding these operations allows us to manipulate matrices without altering their core properties or the systems they represent.

3x3 Matrices

3x3 matrices are frequently encountered in linear algebra, and understanding how to compute their determinants is essential for solving many algebraic problems. To find the determinant of a 3x3 matrix, we commonly use the method of cofactor expansion along a row or a column. This involves breaking down the 3x3 matrix into smaller 2x2 matrices, for which determinants are easier to compute.
The determinant of a 3x3 matrix \[ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \ \end{vmatrix} \]can be calculated as:\[aei + bfg + cdh - ceg - bdi - afh\]
Understanding this process enables quick assessment of the matrix's properties, such as invertibility and linear independence of its rows or columns. Additionally, recognizing patterns and using properties of determinants can further accelerate solutions and error detection during calculations.

Linearity of Determinants

The linearity of determinants plays a pivotal role in understanding how transformations affect the determinant value. This property states that the determinant of a matrix is linear in each of its rows. In simpler terms, if you consider each row of a matrix as a vector, then the determinant is a linear function concerning these vector rows. This means:

• The determinant of the sum of matrices is the sum of their determinants only when those matrices differ by a change confined to one row or one column.


• For constant factors, the determinant of a matrix will be the product of that constant and the determinant of the original matrix. However, be mindful that this applies to the scaling of a single row or column at a time and not the whole matrix immediately.

Linearity extends to combinations of operations, where individual row transformations can be treated independently while considering the constant multipliers applied to them. It's this property that allows the addition of rows multiplied by constants, known as linear combinations, without altering the determinant's value, as observed in problems like determining \(D'\) in terms of \(D\).