Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The rank of the matrix \(A=\left[\begin{array}{rrr}2 & 3 & 4 \\ 3 & 1 & 2 \\\ -1 & 2 & 2\end{array}\right]\) is (A) 1 (B) 2 (C) 3 (D) can't determine

Step-by-Step Solution

Answer

The rank of the matrix is 2.

Step 1: Identify Matrix Rows

The matrix \( A \) is given as: \[A = \begin{bmatrix}2 & 3 & 4 \3 & 1 & 2 \-1 & 2 & 2\end{bmatrix}\]This matrix has three rows: Row 1: \([2, 3, 4]\)Row 2: \([3, 1, 2]\)Row 3: \([-1, 2, 2]\)

Step 2: Determine Row Reduction

We will use row reduction (Gaussian elimination) to transform the matrix into a row-echelon form. Our aim is to identify the number of leading (non-zero) elements in the rows. Let's start by reducing \( A \) with the operations:Transform Row 2: Row 2 = Row 2 - (3/2) * Row 1 \[ \begin{bmatrix} 2 & 3 & 4 \0 & -3.5 & -4 \-1 & 2 & 2 \end{bmatrix} \]Transform Row 3: Row 3 = Row 3 + (1/2) * Row 1 \[ \begin{bmatrix} 2 & 3 & 4 \0 & -3.5 & -4 \0 & 3.5 & 4\end{bmatrix} \]

Step 3: Further Simplify the Matrix

Further simplify to make Row 2 a lead row in the second column:Transform Row 3: Row 3 = Row 3 + Row 2 \[\begin{bmatrix}2 & 3 & 4 \0 & -3.5 & -4 \0 & 0 & 0\end{bmatrix}\]This results in one row being entirely zero.

Step 4: Count Non-zero Rows

Check the number of non-zero rows in the row-echelon form. Here, Row 1 and Row 2 have leading, non-zero elements whereas Row 3 is entirely zero. Thus, there are two leading rows.

Key Concepts

Row Reduction

Row reduction is the process of applying a series of operations to a matrix to transform it into a simpler form. The ultimate goal is to make solving systems of linear equations easier by transforming the matrix into a standard form called row-echelon form. In this exercise, row reduction helps us find the rank of the matrix. Key operations include:

• Swapping two rows.
• Multiplying a row by a nonzero constant.
• Adding or subtracting a multiple of one row to another row.

In the original exercise, we repeatedly performed operations on rows of the matrix to eliminate the elements below the leading diagonals, which are the first non-zero elements in a row from left to right. This makes identifying the rank straightforward by counting the number of non-zero rows.

Gaussian Elimination

Gaussian elimination is a mathematical procedure used to solve systems of linear equations by transforming the matrix into row-echelon form. It systematically uses row operations, like those used in row reduction, to simplify the equations. Its main purpose is to simplify computation, making it easier to analyze and solve. The process typically involves:

• Forward Elimination: Transform the matrix step-by-step to leave it in a triangular form with zeros below each leading coefficient.
• Back Substitution: If necessary, solve for variable values by simplifying from bottom to top, but in calculating the rank, it's not required.

In the exercise, we applied Gaussian elimination to transform the given matrix, helping us see the structure of non-zero elements more clearly for rank determination. Through a sequence of row operations, we reduced the matrix, allowing us to quickly count the leading non-zero rows.

Row-Echelon Form

A matrix is in row-echelon form when all non-zero rows are above any zero rows and the leading coefficient (first non-zero number from the left, also called a pivot) in a non-zero row is to the right of the leading coefficient of the row above it.Characteristics of row-echelon form include:

• All rows consisting entirely of zeros are at the bottom.
• The first non-zero element of each row is in a column to the right of the first non-zero element of the preceding row.
• Each leading entry is the only non-zero entry in its column (only if completely reduced to reduced row-echelon form).

In our exercise, after performing Gaussian elimination, our matrix achieved row-echelon form:\[\begin{bmatrix} 2 & 3 & 4 \ 0 & -3.5 & -4 \ 0 & 0 & 0 \end{bmatrix}\]This form allows easy identification of how many independent rows there are. In this specific case, the matrix rank is determined by the two non-zero rows.