# Triangular Matrix — Upper, Lower, Properties, and Examples

TL;DR

A triangular matrix is a square matrix in which all the entries on one side of the main diagonal are zero — upper triangular when the zeros sit below the diagonal, lower triangular when they sit above. This article covers both types, the strictly-triangular and unit-triangular variants, the key properties, the determinant shortcut, and six worked examples.

## What Is a Triangular Matrix?

A **triangular matrix** is a square matrix in which all the entries either _above_ or _below_ the **main diagonal** are zero. The main diagonal runs from the top-left to the bottom-right.

There are two kinds, named for where the non-zero entries live:

- An **upper triangular matrix** has all zeros _below_ the main diagonal.

- A **lower triangular matrix** has all zeros _above_ the main diagonal.

A triangular matrix must be square, and the main-diagonal entries themselves can be anything — zero or non-zero. A matrix that is both upper _and_ lower triangular at once is a diagonal matrix — every off-diagonal entry is zero.

## What Are the Types of Triangular Matrices?

Beyond upper and lower, two refinements describe the diagonal itself:

- **Strictly triangular.** A strictly upper (or lower) triangular matrix has zeros on the **main diagonal** too.

- **Unit triangular.** A unit triangular matrix has **ones all along the main diagonal**.

So a single triangular matrix can be upper or lower, and within that, strict, unit, or neither.

## What Are the Properties of a Triangular Matrix?

Triangular matrices behave predictably:

- **Determinant = product of the diagonal.**

- **Eigenvalues = the diagonal entries.**

- **Transpose flips the type.**

- **Closed under multiplication and addition.**

- **Invertible only if no diagonal entry is zero.**

- **Inverse stays triangular.**

The determinant and eigenvalue shortcuts are the two properties worth remembering.

## What Is the Determinant of a Triangular Matrix?

For any triangular matrix — upper or lower — the determinant is simply the **product of the main-diagonal entries**. There is no cofactor expansion, no row reduction — just multiply down the diagonal.

## Examples of Triangular Matrix

### Example 1

**Classify A=
\[200530146\]**.

**Final answer:** A is a **lower triangular** matrix.

### Example 2

**Find the determinant of A=
\[472059003\]**.

**Correct:** Multiply the diagonal:

det⁡A=4⋅5⋅3=60.

**Final answer:** det⁡A=60.

### Example 3

**Is A=
\[1800\]** invertible?

**Final answer:** not invertible.

### Example 4

**State the eigenvalues of A=
\[7210−35004\]**.

**Final answer:** the eigenvalues are 7,−3,4.

### Example 5

**Multiply the upper triangular matrices A=
\[2103\] and B=
\[1405\]**.

**Final answer:** AB=
\[213015\] — still upper triangular.

### Example 6

**Solve ;2x+y−z=3,;3y+2z=11,;4z=8;** by back-substitution.

**Final answer:** x=4/3,y=7/3,z=2.

## Why the Triangular Matrix Earns Its Place

Triangular form is the engine behind Gaussian elimination. Where triangular matrices do real work:

- **Solving large linear systems.** 
- **Computing determinants fast.** 
- **Eigenvalue algorithms.** 
- **Computer graphics and statistics.**

## Where Students Trip Up on the Triangular Matrix

### Mistake 1: Expanding the determinant the long way

**The correct way:** Multiply the main-diagonal entries.

### Mistake 2: Mixing up upper and lower

**The correct way:** Upper triangular means zeros _below_ the diagonal; lower triangular means zeros _above_.

### Mistake 3: Forgetting the diagonal can still make it singular

**The correct way:** A triangular matrix is invertible only if **every diagonal entry is non-zero**.

## Key Takeaways

- A **triangular matrix** is a square matrix with all zeros on one side of the main diagonal.
- **Upper** triangular has zeros below the diagonal; **lower** triangular has zeros above it.
- The determinant is the product of the diagonal entries, and the eigenvalues _are_ the diagonal entries.
- A triangular matrix is invertible only when no diagonal entry is zero.