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

Question

The value of determinant $$ \left|\begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+a & 1 \\ 1 & 1 & 1+a \end{array}\right| \text { is: } $$ (a) \(a^{3}\left(1-\frac{2}{a}\right)\) (b) \(a^{3}\left(1+\frac{3}{a}\right)\) (c) \(a^{3}\left(1-\frac{3}{a}\right)\) (d) \(a^{3}\left(1+\frac{2}{a}\right)\)

Step-by-Step Solution

Answer

The determinant value simplifies to zero after row reductions.

Step 1: Setup the Determinant

We are given a determinant \( D \): \[ D = \begin{vmatrix} 1+a & 1 & 1 \ 1 & 1+a & 1 \ 1 & 1 & 1+a \end{vmatrix} \] Our goal is to simplify and calculate it.

Step 2: Simplify the Determinant

Subtract the first column from the second and third columns. This simplification can make the determinant easier to solve: \[ D = \begin{vmatrix} 1+a & 1 - (1+a) & 1 - (1+a) \ 1 & 1 - 1 & 1 - 1 \ 1 & 1 - 1 & 1 - 1 \end{vmatrix} = \begin{vmatrix} 1+a & -a & -a \ 1 & 0 & 0 \ 1 & 0 & 0 \end{vmatrix} \]

Step 3: Calculate the Determinant

Notice that the new matrix has two identical rows (second and third), making the determinant equal to zero. Therefore, \( D = 0 \).

Step 4: Verify the Options

Compare the result (which is zero) to the provided options. Observe that substituting \( a = 0 \) in all options should give a result of zero. However, none of the options explicitly simplifies to zero at any point, indicating a mistake or a trick question.

Key Concepts

Matrix Row Operations

Matrix row operations are crucial when working with determinants because they can simplify the calculation process. These operations include:

• Swapping two rows
• Multiplying a row by a scalar
• Adding or subtracting a multiple of one row to/from another row

These operations can transform the matrix into a form that is easier to handle. In the context of determinants, the key is to remember that some operations might affect the determinant's value, while others won't. For instance, swapping rows changes the sign of the determinant, but simply adding a multiple of one row to another does not change its value.
In our step-by-step solution, we used a specific row operation: subtracting the first column from the second and third columns. This operation was used to introduce zeros into the matrix, thereby making the problem simpler to solve. However, it's crucial to apply these operations correctly to avoid errors.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces and through matrices. An essential part of linear algebra is the study of determinants, which helps in understanding the properties of matrices.
Determinants provide important information about matrices. They can indicate whether a set of vectors is linearly independent and whether a matrix is invertible. These properties are foundational for solving systems of linear equations and performing matrix transformations. In our exercise, understanding the determinant's properties revealed that the matrix had a zero determinant when simplified, implying linear dependence among the matrix's rows.
Working knowledge of linear algebra principles aids in recognizing patterns within matrices, applying appropriate transformations, and predicting the effects of operations on determinants.

Zero Determinant Property

The zero determinant property is a fundamental principle in linear algebra. This property states that if a matrix has rows or columns that are linearly dependent, then its determinant is zero.
Such situations occur when:

• Two rows or columns in a matrix are identical
• One row or column is a linear combination of others

In our exercise, after performing a row operation, we found two identical rows in the matrix, leading directly to a zero determinant.
This property is significant because it indicates certain matrix characteristics. Specifically, a zero determinant means that a matrix:

• Is not invertible
• Represents a system of equations with no unique solutions
• Reflects linear dependence among vectors represented by its rows or columns

Understanding the zero determinant property is key to assessing matrix behavior and solving linear algebra problems effectively.