Q.Evaluate $\Delta = \begin{vmatrix} 0 & \cos x & -\sin x \\ -\cos x & 0 & \cos y \\ \sin x & -\cos y & 0 \end{vmatrix}$.

What is an Orthogonal Matrix? The Intuition

Imagine you have a set of coordinate axes — the usual $x$, $y$, and $z$ directions. Now suppose you rotate the whole space. The axes change direction, but something important stays the same: lengths of vectors, angles between them, and the fact that the new axes are still perpendicular (orthogonal) to each other.

An orthogonal matrix is the mathematical tool that describes exactly this kind of transformation — a rotation (or a reflection) that preserves distances and angles. It's a square matrix whose columns are mutually perpendicular unit vectors.

The Precise Definition

A square matrix $Q$ of size $n \times n$ is called orthogonal if its transpose equals its inverse:

where $I$ is the identity matrix. Equivalently:

This single equation captures everything. Let's see why.

What This Property Actually Means

1. Columns are orthonormal

Write $Q$ in terms of its column vectors: $Q = [\mathbf{c}_1 \; \mathbf{c}_2 \; \cdots \; \mathbf{c}_n]$.

When you compute $Q^T Q$, the $(i,j)$ entry is the dot product $\mathbf{c}_i \cdot \mathbf{c}_j$. The equation $Q^T Q = I$ says:

• For $i = j$: $\mathbf{c}_i \cdot \mathbf{c}_i = 1$ — each column is a unit vector.
• For $i \neq j$: $\mathbf{c}_i \cdot \mathbf{c}_j = 0$ — different columns are perpendicular.

So the columns form an orthonormal basis for $\mathbb{R}^n$.

2. Rows are also orthonormal

Exactly the same reasoning applied to $Q Q^T = I$ shows that the rows of $Q$ also form an orthonormal set. This is a special symmetry — if columns are orthonormal, rows automatically are too.

3. Lengths are preserved

For any vector $\mathbf{x}$, the transformed vector $Q\mathbf{x}$ has the same length:

This is why orthogonal matrices are called isometries — they preserve distances.

4. Angles are preserved

The dot product is also preserved:

Since angles are defined through dot products, angles between vectors remain unchanged.

A Simple 2×2 Example

The rotation matrix by angle $\theta$:

Check: $Q^T = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$, and

The columns are $(\cos\theta, \sin\theta)$ and $(-\sin\theta, \cos\theta)$ — both unit vectors, and their dot product is $-\cos\theta\sin\theta + \sin\theta\cos\theta = 0$.

Why This Matters in Exams

The orthogonal matrix property appears in:

• Linear algebra: diagonalizing symmetric matrices (they have orthogonal eigenvectors)
• Coordinate geometry: changing between orthonormal bases
• Physics: representing rotations in 3D space
• Statistics: principal component analysis (PCA)

Common Mistake to Avoid

The Bottom Line

An orthogonal matrix is a square matrix whose columns (and rows) form an orthonormal set. It preserves lengths and angles, and its inverse is simply its transpose.