Problem 18 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Without expanding the determinant at any stage, show that $$ \left|\begin{array}{ccc} x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1 \end{array}\right|=A x+B $$

Step-by-Step Solution

Answer

Determinant is zero due to linear dependence.

Step 1: Identify Linear Dependence

Examine the rows of the determinant for linear dependence using the properties of determinants. Notice this in the second and third rows: if we subtract a multiple of one row from another, we identify dependence more clearly. Specifically, the third row looks like a linear combination of the first two.

Step 2: Row Operations to Identify Zero Rows

Perform row operations to detect a zero row, which indicates a linear relationship among the rows. Subtract the first row multiplied by certain integers from the second and third rows under observation of patterns.

Step 3: Simplify Using Properties of Determinants

Notice that after performing the row operations, one of the rows becomes a zero row (indicating linear dependence among the rows). Consequently, the entire determinant is zero. When represented as a polynomial, it confirms that the constant term A and linear term B satisfy that dependency without further expansion.

Key Concepts

Linear Dependence

Linear dependence is a critical concept in understanding matrices and determinants. In simple terms, a set of vectors (or matrix rows) is said to be linearly dependent if one vector in the set can be defined as a linear combination of the others. This concept is important because it helps identify cases where rows or columns of a matrix don't add new information, impacting the matrix's determinant.

In the context of determinants, when rows of a matrix are linearly dependent, it means that one or more rows can be expressed as a sum of the others, usually times some coefficients. This implies that the determinant of the matrix is zero, highlighting redundancy in the system.

To identify linear dependence, students can:

• Look for obvious multiple relationships between rows.
• Use row operations to transform the matrix, making dependence clearer.
• Check if linear combinations result in a row of zeros.

Understanding linear dependence ensures a deeper comprehension of how matrix transformations affect the determinant.

Row Operations

Row operations are essential tools in matrix algebra, and they simplify the process of computing determinants and solving systems of linear equations. These operations include swapping rows, multiplying a row by a scalar, and adding or subtracting rows.

Using row operations, students can simplify matrices, particularly when assessing determinants. Performing row operations does not change the determinant's value except in the swapping of rows, which changes the sign of the determinant but doesn't affect its absolute value.

For the problem at hand, students can use row operations to:

• Transform parts of the matrix, revealing zero rows if linear dependence is present.
• Reduce complexity by making numbers smaller and easier to work with.
• Detect identical or multiple rows, which simplifies calculations and insights.

Through careful application, row operations unveil underlying relationships between rows, aiding in identifying linear dependence and simplifying the matrix structure.

Properties of Determinants

The properties of determinants are pivotal in comprehending how they behave under various matrix manipulations. Some essential properties include:

• If any two rows are identical, the determinant is zero, showing linear dependence.
• Swapping two rows changes the determinant's sign.
• Scaling a row by a factor multiplies the determinant by that factor.
• Adding a multiple of one row to another leaves the determinant unchanged.

In the exercise, these properties help determine the matrix's determinant without fully expanding it. When a zero row emerges after certain row operations, it confirms the linear dependence, achieving a determinant of zero.

Recognizing these properties allows students to:

• Avoid full determinant expansion by leveraging the structural features of the matrix.
• Understand how transformations influence the determinant value.
• Develop strategies for efficiently checking determinant values in complex matrices.

Mastering these properties equips students to handle matrix algebra with greater ease and confidence.