A nonsingular matrix and several systems with the matrix as coefficient matrix. Always a unique solution by Theorem NSMUS.
{{{ A = matrix(QQ, [[4,3,3,3,-1], [-1,7,-4,7,2], [-1,-1,1,3,4], [-4,2,-4,7,2], [7,-4,1,3,-2]]) A }}}Check that $A$ is nonsingular with Theorem NMRRI.
{{{ A.rref() }}}Try negating the next command with not.
{{{ A.is_singular() }}} {{{ b = vector(QQ, [30,12,13,11,38]) temp = A.augment(b) temp.rref() }}} {{{ b = vector(QQ, [-7,7,9,5,-20]) temp = A.augment(b) temp.rref() }}}A very small null space (called a ”right kernel” in Sage).
{{{ NA = A.right_kernel() NA.list() }}}
Now a singular matrix.
{{{ B = matrix(QQ, [[7,-1,-12,21,-8], [-3,3,0,-9,6], [3,3,-12,9,0], [2,3,-10,6,1], [-2,2,0,-6,4]]) B }}}Check that $B$ is nonsingular with Theorem NMRRI.
{{{ B.rref() }}} {{{ B.is_singular() }}}A system with infinitely many solutions.
{{{ c = vector(QQ, [17,-15,-3,-5,-10]) temp = B.augment(c) temp.rref() }}}A system with no solutions. Then ask which choices for the vector c lead to consistent systems?
{{{ c = vector(QQ, [-3,1,-2,1,2]) temp = B.augment(c) temp.rref() }}}A very large null space (called a ”right kernel” in Sage).
{{{ NB = B.right_kernel() NB.list() }}}