Matrix Rank & System Consistency: JEE Main Maths

Concept Overview

This question tests your ability to find the rank of a matrix using elementary row operations to transform it into row echelon form. It also assesses your understanding of how the ranks of the coefficient matrix and the augmented matrix relate to the consistency (existence of solutions) and uniqueness of solutions for a system of linear equations, as described by the Rouché-Capelli theorem.

Step 1: Write the system of linear equations in matrix form. A system of linear equations can be represented as $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix. For the given system: $x + y + z = 6$ $x + 2y + 3z = 14$ $2x + 5y + z = 18$

The matrix form is:

Here, $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 5 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 6 \\ 14 \\ 18 \end{pmatrix}$.

Step 2: Form the augmented matrix. The augmented matrix $[A|B]$ combines the coefficient matrix $A$ and the constant matrix $B$. This matrix is used to perform row operations to determine the rank and consistency. The augmented matrix is:

Step 3: Reduce the augmented matrix to row echelon form. We use elementary row operations (swapping rows, multiplying a row by a non-zero scalar, adding a multiple of one row to another) to transform the matrix into row echelon form. In row echelon form, the first non-zero element (pivot) in each row is to the right of the pivot in the row above it, and all entries below the pivots are zero. Let $R_i$ denote the $i$-th row. Apply $R_2 \leftarrow R_2 - R_1$ and $R_3 \leftarrow R_3 - 2R_1$:

Apply $R_3 \leftarrow R_3 - 3R_2$:

This matrix is now in row echelon form.

Step 4: Determine the rank of the coefficient matrix ($A$) and the augmented matrix ($[A|B]$). The rank of a matrix is the number of non-zero rows in its row echelon form. For the coefficient matrix $A$, its row echelon form is obtained by considering only the first three columns of the row echelon form of $[A|B]$:

There are 3 non-zero rows. So, $rank(A) = 3$.

For the augmented matrix $[A|B]$, its row echelon form is:

There are 3 non-zero rows. So, $rank([A|B]) = 3$.

Step 5: Determine the consistency of the system using the Rouché-Capelli theorem. The Rouché-Capelli theorem states:

1. If $rank(A) \neq rank([A|B])$, the system is inconsistent (no solution).
2. If $rank(A) = rank([A|B]) = r$: a. If $r = n$ (where $n$ is the number of variables), the system has a unique solution. b. If $r < n$, the system has infinitely many solutions.

In this case, $rank(A) = 3$ and $rank([A|B]) = 3$. The number of variables ($x, y, z$) is $n=3$. Since $rank(A) = rank([A|B]) = 3$ and $n=3$, the system has a unique solution.

Step 6: (Optional, but good for completeness) Find the solution. From the row echelon form of the augmented matrix: $-7z = -18 \implies z = \frac{18}{7}$ $y + 2z = 8 \implies y = 8 - 2z = 8 - 2\left(\frac{18}{7}\right) = 8 - \frac{36}{7} = \frac{56 - 36}{7} = \frac{20}{7}$ $x + y + z = 6 \implies x = 6 - y - z = 6 - \frac{20}{7} - \frac{18}{7} = 6 - \frac{38}{7} = \frac{42 - 38}{7} = \frac{4}{7}$ The unique solution is $x = \frac{4}{7}, y = \frac{20}{7}, z = \frac{18}{7}$.

Key Takeaways:

• The rank of a matrix is the number of non-zero rows in its row echelon form.
• Elementary row operations do not change the rank of a matrix.
• The Rouché-Capelli theorem relates the ranks of the coefficient and augmented matrices to the existence and number of solutions for a system of linear equations.
• A system is consistent if and only if $rank(A) = rank([A|B])$.

Answer: The system has a unique solution because $rank(A) = rank([A|B]) = 3$, which is equal to the number of variables.