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.