Section 7.1 Determinants
The determinant of a square matrix \(A\) is a number \(det(A)\) defined by these properties:
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Adding a multiple of any row in \(A\) to another row does not change \(det(A)\text{.}\)
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Multiplying any row of \(A\) by a scalar \(k\) multiplies \(det(A)\) by \(k\text{.}\)
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Swapping two rows of \(A\) multiplies its determinant by \(-1\text{.}\)
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The determinant of the identity matrix is \(1\text{.}\)
In Sage, the determinant of a matrix can be computed using the
det() method:
A matrix is singular if its determinant is zero. In Sage, to check whether a matrix is singular or not the method
is_singular() can be used. Note that just as with the determinant, this method is only applicable to square matrices.
The rank of a matrix is the largest dimension of any of its square submatrix with a non-zero determinant. In Sage the rank of a matrix is computed with the
rank() method:
Alternatively, the rank is also the number of pivots in the row reduction of the matrix. In Sage, the row reduction can be performed with the
rref() method, which returns a tuple containing the reduced row echelon form of the matrix and a list of pivot columns. Another way to get the rank of a matrix is to count how may pivots the matrix has.
