Rank of a Matrix

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter extends earlier matrix and determinant work to three genuinely practical tools: finding the rank of a matrix, using rank to test whether a system of linear equations can be solved at all, solving a three-variable system by Cramer's Rule, and modelling how probabilities shift over time using a transition probability matrix.

What is the rank of a matrix?

For a matrix $A$, a minor of order $r$ is the determinant of any $r\times r$ square sub-matrix obtained by deleting rows/columns of $A$. The rank of $A$, written $\rho(A)$, is the order of the largest non-zero minor — equivalently, the number of linearly independent rows (or columns) of $A$.

In practice, rank is found by using elementary row operations to reduce $A$ to echelon form (a staircase pattern of leading non-zero entries, with only zeros below each leading entry) — the rank then equals the number of non-zero rows remaining.

Example: reducing a matrix to echelon form

For $A=\begin{pmatrix}1&2&3\\2&3&4\\3&5&7\end{pmatrix}$, apply $R_2\to R_2-2R_1$ and $R_3\to R_3-3R_1$:

Then $R_3\to R_3-R_2$ gives

Two non-zero rows remain, so $\rho(A)=2$.

This echelon-form method is the standard, numerically stable way rank is computed, and it is exactly what the next section uses to test whether a system of equations can be solved at all — a genuinely practical business-mathematics application, matching how CBSE/NCERT mathematics and applied-mathematics courses across India treat rank and linear-system consistency.