Determinant, Magnitude and Inverse Operators

Keystroke: |

|v| or |M|

Returns the magnitude (norm) of a vector, the absolute value of a scalar, or the determinant of a matrix. This operator tries to choose the correct functional meaning based on the value of its argument. When it is unable to determine the argument (for example, when it is used as part of a function definition), it defaults to the norm or absolute value definition. To explicitly set the operator to take the determinant, right-click on it, and choose determinant from the menu.

Keystroke: ^−1

M^−1

Returns the inverse of a matrix. If M does not have an inverse, an appropriate error message appears. To compute the determinant or inverse, Mathcad performs an LU decomposition of the matrix. The BLAS/LAPACK libraries (http://www.intel.com/software/products/mkl/features/lin_alg.htm) from Intel are used.

Operands:

• v is a column vector or a scalar.

• M is a dimensionless square matrix.

Notes:

• |v| is the square root of the sum of the squared magnitudes of the elements in v. To take the magnitude of each element in a matrix or vector, place |M| between the editing lines, and apply the vectorize operator, [Ctrl]−.
• All square matrices that have a determinant are invertible. However, if a matrix has a determinant near zero, then the typical algorithms that compute inverses become unstable and fail to work. Mathcad uses LU decomposition, which becomes unstable when a matrix is:
  • singular - its determinant is equal to zero,
  • nearly singular - a linear combination of some rows nearly reproduces another row, as indicated by a high condition number,
  • or ill-conditioned - the ratio of the largest magnitude eigenvalue to the smallest is greater than the precision of floating point numbers.

In these cases, matrix inversion may give an error, or it may give results that are not a genuine inverse matrix, that is, M·M^-1 may not be equal to the identity matrix. When matrix inversion fails, you can try the geninv function. If you get invalid results, you can check the strict singularity check box in Worksheet Options, which uses a slower algorithm that rejects matrices that are nearly singular and gives an error.

• You can also use these operators to find symbolic determinants, magnitudes, and inverses, using the live symbolic operator.

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