Revision #2578 → #3163 · back to history
modifiedReduced row echelon form37cf991fe4a5
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| anchors | [{"section":"(Lead)","snippet":"a matrix can always be transformed into reduced row echelon form"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}1&3&1&9\\\\1&1&-1&1\\\\3&11&5&35\\end{bmatrix}}\\to {\\begin{bmatrix}1&3&1&9\\\\0&-2&-2&-8\\\\0&2&2&8\\end{bmatrix}}\\to {\\begin{bmatrix}1&3&1&9\\\\0&-2&-2&-8\\\\0&0&0&0\\end{bmatrix}}\\to {\\begin{bmatrix}1&0&-2&-3\\\\0&1&1&4\\\\0&0&0&0\\end{bmatrix}}}"}] | — |
modifiedUniqueness of reduced row echelon form5828570cbe8b
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| anchors | [{"section":"(Lead)","snippet":"This final form is unique"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}1&3&1&9\\\\1&1&-1&1\\\\3&11&5&35\\end{bmatrix}}\\to {\\begin{bmatrix}1&3&1&9\\\\0&-2&-2&-8\\\\0&2&2&8\\end{bmatrix}}\\to {\\begin{bmatrix}1&3&1&9\\\\0&-2&-2&-8\\\\0&0&0&0\\end{bmatrix}}\\to {\\begin{bmatrix}1&0&-2&-3\\\\0&1&1&4\\\\0&0&0&0\\end{bmatrix}}}"}] | — |
addedElementary matrix6d078cfa0183
modifiedMatrix in row echelon form72ce5991f9ab
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| anchors | [{"section":"Echelon form","snippet":"the following matrix is in row echelon form"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}0&\\color {red}{\\mathbf {2} }&1&-1\\\\0&0&\\color {red}{\\mathbf {3} }&1\\\\0&0&0&0\\end{bmatrix}}.}"}] | — |
modifiedSolving a system by row reduction95e48b93955c
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| anchors | [{"section":"Example of the algorithm","snippet":"Suppose the goal is to find and describe the set of solutions"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{alignedat}{4}2x&{}+{}&y&{}-{}&z&{}={}&8&\\qquad (L_{1})\\\\-3x&{}-{}&y&{}+{}&2z&{}={}&-11&\\qquad (L_{2})\\\\-2x&{}+{}&y&{}+{}&2z&{}={}&-3&\\qquad (L_{3})\\end{alignedat}}}"}] | — |
modifiedDeterminant via row echelon formed4e4ea81dec
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| anchors | [{"section":"Computing determinants","snippet":"the determinant of A is the quotient by d of the product of the elements of the diagonal of B"},{"type":"math_alttext","value":"{\\displaystyle \\det(A)={\\frac {\\prod \\operatorname {diag} (B)}{d}}.}"}] | — |
modifiedComputing a matrix inversec38e38d75c30
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| anchors | [{"section":"Finding the inverse of a matrix","snippet":"To find the inverse of this matrix, one takes the following matrix augmented by the identity"},{"type":"math_alttext","value":"{\\displaystyle [A|I]=\\left[{\\begin{array}{ccc|ccc}2&-1&0&1&0&0\\\\-1&2&-1&0&1&0\\\\0&-1&2&0&0&1\\end{array}}\\right].}"}] | — |
modifiedRank and basis from echelon form22cb43804b7d
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| anchors | [{"section":"Computing ranks and bases","snippet":"the rank of A is 5, since there are 5 nonzero rows in T"},{"type":"math_alttext","value":"{\\displaystyle T={\\begin{bmatrix}a&*&*&*&*&*&*&*&*\\\\0&0&b&*&*&*&*&*&*\\\\0&0&0&c&*&*&*&*&*\\\\0&0&0&0&0&0&d&*&*\\\\0&0&0&0&0&0&0&0&e\\\\0&0&0&0&0&0&0&0&0\\end{bmatrix}},}"}] | — |
addedHadamard bound for Bareiss entries66bf23e7328a
addedPartial pivoting846c57c8396e