Matrix Type Checker: Symmetric, Skew-Symmetric, Diagonal, Triangular, Orthogonal, Idempotent, Nilpotent, Singular

Find out what kind of matrix you have.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Type in the nine numbers of the matrix.

The calculator finds the determinant, the trace and the rank, and tests the matrix against each definition: zero, identity, scalar, diagonal, upper and lower triangular, symmetric, skew-symmetric, orthogonal, idempotent, involutory and nilpotent.

It also says whether the matrix is singular (determinant zero) or non-singular.

Formulas used

  • A^T = A\ \ (\text{symmetric})
  • A^T = -A\ \ (\text{skew})
  • A^2 = A\ \ (\text{idempotent})
  • AA^T = I\ \ (\text{orthogonal})
  • A^k = O\ \ (\text{nilpotent})

Worked example

With the default numbers in the calculator above:

  1. Determinant: -7.00
  2. Trace: the sum of the diagonal: 13.00
  3. Rank: 3
  4. Zero matrix: every element is zero: No
  5. Identity matrix: ones on the diagonal and zeros elsewhere: No
  6. Scalar matrix: a diagonal matrix with all the diagonal elements equal: No
  7. Diagonal matrix: every element off the diagonal is zero: No
  8. Upper triangular: every element below the diagonal is zero: No
  9. Lower triangular: every element above the diagonal is zero: No
  10. Symmetric: the matrix equals its transpose: Yes
  11. Skew-symmetric: the transpose is the negative of the matrix: No
  12. Orthogonal: the transpose is the inverse: No

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Row 1, column 1 0; Row 1, column 2 2; Row 1, column 3 -3; Row 2, column 1 -2; Row 2, column 2 0; Row 2, column 3 4; Row 3, column 1 3; Row 3, column 2 -4; Row 3, column 3 0.

  1. Determinant: 0.00
  2. Trace: the sum of the diagonal: 0.00
  3. Rank: 2
  4. Zero matrix: every element is zero: No
  5. Identity matrix: ones on the diagonal and zeros elsewhere: No
  6. Scalar matrix: a diagonal matrix with all the diagonal elements equal: No
  7. Diagonal matrix: every element off the diagonal is zero: No
  8. Upper triangular: every element below the diagonal is zero: No
  9. Lower triangular: every element above the diagonal is zero: No
  10. Symmetric: the matrix equals its transpose: No
  11. Skew-symmetric: the transpose is the negative of the matrix: Yes
  12. Orthogonal: the transpose is the inverse: No

Common questions

What is a singular matrix?

A square matrix whose determinant is zero. It has no inverse.

What is a nilpotent matrix?

One that becomes the zero matrix when it is multiplied by itself enough times.

Can a matrix be more than one type?

Yes. The identity matrix is diagonal, scalar, symmetric, triangular, orthogonal and idempotent at once.

What is an orthogonal matrix?

A matrix whose transpose is its inverse. Rotation matrices are orthogonal.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Strang, Linear Algebra and Its Applications; Kreyszig, Advanced Engineering Mathematics (matrices).

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