Find the rank of a 3×3 matrix using row echelon form

Question

Find the rank of the matrix:

A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 8 \end{pmatrix}

(JEE Main 2022, similar pattern)

Solution — Step by Step

Start with the matrix and perform $R_2 \to R_2 - 2R_1$ and $R_3 \to R_3 - 3R_1$ :

\begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -1 \\ 0 & -1 & -1 \end{pmatrix}

Apply $R_3 \to R_3 - R_2$ :

\begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -1 \\ 0 & 0 & 0 \end{pmatrix}

This is the row echelon form.

The row echelon form has 2 non-zero rows.

\boxed{\text{Rank}(A) = 2}

Why This Works

The rank of a matrix equals the number of non-zero rows in its row echelon form. Row operations don’t change the rank because they don’t change the row space — they just rewrite the same set of linear combinations in a simpler form.

The third row becoming all zeros means $R_3$ was a linear combination of $R_1$ and $R_2$ . Indeed, $(3, 5, 8) = (1, 2, 3) + (2, 3, 5)$ . So only 2 of the 3 rows are linearly independent.

An equivalent check: if $|A| = 0$ (which it is here, since a row of zeros gives determinant 0), then the rank is strictly less than 3. We then check if any $2 \times 2$ minor is non-zero to confirm rank = 2.

Alternative Method — Determinant approach

Compute $|A|$ : expand and get $|A| = 0$ (you can verify: the third row is the sum of the first two).

Since $|A| = 0$ , rank $< 3$ . Now check $2 \times 2$ minors. Take the top-left $2 \times 2$ : $\begin{vmatrix} 1 & 2 \\ 2 & 3 \end{vmatrix} = 3 - 4 = -1 \neq 0$ .

Since a non-zero $2 \times 2$ minor exists, rank $\geq 2$ . Combined: rank = 2.

For JEE MCQs, the determinant method is faster. Check $|A| = 0$ (rank < 3), then check one $2 \times 2$ minor (if non-zero, rank = 2). The row echelon method is more systematic for larger matrices or when you need to find the actual independent rows.

Common Mistake

Students sometimes conclude that rank = 0 when they see a row of zeros, confusing “one zero row” with “rank is zero.” Rank equals the number of non-zero rows, not zero rows. A $3 \times 3$ matrix with one zero row in echelon form has rank 2, not 0 or 1. Rank = 0 only for the zero matrix (all entries zero).

Question

Solution — Step by Step

Why This Works

Alternative Method — Determinant approach

Common Mistake

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