Browse examples
Example Solutions for

Step-by-Step Linear Algebra

Matrix Properties

Compute properties of a matrix step by step:

{{1, -1}, {2, 3}}
{{2,4,4},{1,2,2},{0,7,3}}
More examples
Matrix Arithmetic

See the steps to multiply matrices:

{{1, 2}, {3, 4}} . {{-1, 1}, {0, 2}}

Invert a matrix one step at a time:

inverse {{1, 1, 2}, {-1, 2, 2}, {3, 2, 3}}
More examples
Determinants

Find the determinant step by step with various methods:

determinant of {{1,2}, {-1, 2}}
determinant of {{1,2,1}, {1,1,0}, {0,1,1}}
More examples
Row Reduction

Write a matrix in reduced row echelon form one step at a time:

{{1,1,5},{1,-1,1}} row reduce
reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}}
More examples
Rank & Nullity

Find the rank of a matrix:

rank {{1, 2, 1}, {-2, -3, 1}, {3, 5, 0}}

Find the nullity of a matrix:

nullity {{1, 2}, {3, 4}, {5, 6}, {7, 8}}
More examples
Null Space

Find the null space of a matrix:

null space {{1, 3, 3}, {-3, -5, -3}, {3, 3, 0}}
kernel {{0, 1, 0},{-1, 0, 2},{0, -1, 0}, {0, 0, -1}}
More examples
Eigenvalues & Eigenvectors

Compute eigenvalues and eigenvectors step by step:

eigenvalues {{3,-1},{0,2}}
eigenvectors {{7,0,-3},{-9,-2,3},{18,0,-8}}
More examples
Characteristic Polynomial

Find the characteristic polynomial of a matrix:

characteristic polynomial {{1, 2}, {-1, 4}}
char poly {{10,-35,50,-24},{1,0,0,0},{0,1,0,0},{0,0,1,0}}
More examples
Vector Arithmetic

See how to compute a dot product:

(4, 1) . (-2, 3)

Compute a cross product step by step:

(1,2,3)x(3,4,5)
More examples
Vector Norm

Calculate a vector norm or length:

vector length {2,3,4}
norm of (y, Sqrt(3), 3x)
More examples
Linear Independence

Determine the linear independence of vectors step by step:

linear independence {1,0,0},{2,0,0},{0,4,5}
linear independence of (2,x,6),(5y,1,0),(0,0,1)
More examples