Worked examples that make the tools easier to use.

Example 1, row reduce a 3×3 matrix

Suppose you want to reduce the matrix below to reduced row echelon form.

Result

1. Subtract 2 times row 1 from row 2.
2. Subtract 3 times row 1 from row 3.
3. Swap the second and third rows.
4. Scale the new second row to make the pivot 1.

Example 2, determinant check before finding an inverse

Before trying to invert a matrix, check whether its determinant is zero.

Determinant: 10

Because the determinant is not zero, the matrix is invertible. That makes it worth moving to the inverse calculator instead of wasting time on a singular matrix.

Example 3, inverse of a 2×2 matrix

For a small invertible matrix, the inverse gives you the matrix that returns the identity when multiplied back.

This gives you a fast answer and a simple path to double-check it with the matching calculator.

Example 4, add two 2×2 matrices

Matrix addition works entry by entry, so each value in the result comes from the matching positions in both matrices.

A + B:

This is a quick check when you want to verify the result of a simple matrix operation before moving on to something larger.

Example 5, trace of a 3×3 matrix

The trace is the sum of the main diagonal. It gives you one quick summary value without needing the full determinant or inverse.

Trace: 3 + 5 + (-2) = 6

That makes trace a fast check when you want a compact value from the matrix before moving on to a bigger calculation.

Example 6, rank of a 3×3 matrix

The rank tells you how many independent rows or columns the matrix really has after row reduction.

Rank: 2

One row depends on another here, so the matrix is not full rank. That makes rank a fast dependency check before solving bigger linear algebra problems.

Example 7, identity matrix for an inverse check

The identity matrix is the target result when a matrix is multiplied by its inverse.

If your inverse check works, the final multiplication should match this pattern.