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Matrix Column Space Calculator

Enter the nine entries of your 3×3 matrix.

How it is calculated

Enter the entries aij (row i, column j) of a 3×3 matrix; all nine must be filled in, otherwise there is no result. The tool row-reduces a copy of the matrix with Gauss-Jordan elimination, picking the largest available entry (by absolute value) in each column as pivot and treating anything below 10-9 as zero. A column that gets a pivot is a pivot column. The rank is the number of pivots, and the matching columns of your original matrix (not the reduced one) form a basis of the column space. An all-zero matrix has rank 0 and its column space is just {0}. The determinant is computed from the original matrix by cofactor expansion along the first row and shown with four decimals. Rank 3 means a non-zero determinant and an invertible matrix.

Worked example

Take the rows (1, 2, 3), (2, 4, 6), (1, 0, 1). In column 1 the largest entry is 2, so row 2 is the pivot row; after scaling and clearing, row 1 becomes all zeros and row 3 becomes (0, −2, −2). Column 2 therefore gets a pivot from row 3. Column 3 has nothing left below the two pivots, so no third pivot appears. The rank is 2, the tool prints "Basis = pivot columns c1, c2 (dim 2 subspace of R^3)", and the determinant is 0.0000. This fits c3 = c1 + c2: (1, 2, 1) + (2, 4, 0) = (3, 6, 1).

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Common mistakes

  • Taking the pivot columns from the reduced matrix. The basis uses the original columns c1, c2 and so on, not the columns of the echelon form.
  • Entering rows as columns. The cell aij means row i, column j; a transposed entry changes which columns are listed, even though the rank stays the same.
  • Assuming a non-zero rank means the matrix is invertible. Only rank 3 (determinant not zero) is invertible for a 3×3 matrix; rank 2 or 1 gives determinant 0.
  • Leaving a cell blank and expecting zero. All nine entries are required, so type 0 explicitly or no result appears.
  • Expecting a unique basis. The columns listed are one valid basis; for example the matrix with rows (0, 1, 2), (0, 3, 4), (0, 5, 6) lists c2 and c3 because column 1 is all zeros.

About this calculator

The column space of a matrix is the set of all linear combinations of its columns — geometrically, the subspace that Ax can actually reach as x ranges over all vectors. Its dimension equals the matrix's rank, and a basis for it is given by the original columns of the matrix that correspond to the pivot positions found during Gaussian elimination to row-echelon form (non-pivot columns are redundant, being linear combinations of the pivot ones).

For a 3×3 matrix, this calculator reduces the matrix to echelon form to identify the pivot columns, reports the rank and a basis for the column space, and computes the determinant. The column space determines whether a linear system Ax = b has a solution for a given b (it does exactly when b lies in the column space), which is why this is a core tool in solving systems of equations, checking invertibility, and understanding linear transformations in engineering and computer graphics.

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  • Editorial team

    A single entry can change the rank. The matrix with rows 1 2 3, 4 5 6, 7 8 10 has rank 3 and determinant -3.0000, but replace the 10 with a 9 and it drops to rank 2 with determinant 0.0000. Near-singular matrices from measured data may hide this, because the tool treats values below a tiny threshold as zero. Try your own matrix with small changes to see how fragile it is. Would a 4 by 4 version help you?

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