Q.A matrix which is not a square matrix is called a _________ matrix.

What is a Square Matrix? The Intuition First

Arrange numbers in a grid. If the grid has the same number of rows as columns — like a chessboard (8×8) or a tic-tac-toe board (3×3) — that's a square matrix. The name comes from the shape: it looks like a square, not a rectangle.

Why care? Square matrices are the only ones that can be multiplied by themselves, have a determinant, or be inverted — the workhorses behind solving systems of equations, computer graphics, and quantum mechanics.

The Precise Definition

That is, the number of rows equals the number of columns. If a matrix has $m$ rows and $n$ columns its order is $m \times n$; for a square matrix we write the order as $n \times n$, or say it's "of order $n$".

Example: $B = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 5 & 7 \\ -4 & 1 & 8 \end{bmatrix}$ is a $3 \times 3$ square matrix.

Counterexample: $C = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$ is $3 \times 2$ — 3 rows, 2 columns. Not square, but rectangular.

Key Properties You'll Meet

Once a matrix is square, several features become possible:

• Main diagonal: the entries $a_{11}, a_{22}, \dots, a_{nn}$ (top-left to bottom-right).
• Determinant: a single number $\det(A)$ (or $|A|$); rectangular matrices don't have one.
• Inverse: a square $A$ may have an inverse $A^{-1}$ with $A A^{-1} = I_n$. Only square matrices can be invertible.
• Trace: the sum of the diagonal entries, $\text{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}$.

Special Square Matrices You Should Know