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Last updated 4:23 AM on 9/21/26
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42 Terms
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The pivot of a row in a matrix is…….
The left-most non zero entry in that row
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A matrix in row echelon form if…..
"1) all rows of zeros [if there are any] are at the bottom
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2) the pivot in every non-zero row is in a column to the right of the pivot in the row above it"
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An example of a matrix in row echelon form….
“Staircase pattern”
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A matrix is in reduced row echelon form if….
"1) in row echelon form
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2) all of the pivots are equal to 1
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3) each pivot is the only non-zero entry in its column"
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An example of a matrix in reduced row echelon form….
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xi is a pivot variable for the system if….
"Letting C be a coefficient matrix,
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rref(C) has a pivot in the ith column"
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xi is a non-pivot variable of the system if…
"Letting C be a coefficient matrix,
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rref(C) doesn’t have a pivot in the ith column"
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Example of a pivot variable and non pivot variable:
"System:
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x + 2y = 2
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z = 3
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pivot variable:
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x, z
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non- pivot variable:
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y"
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Lemma: Pivot Parameterization
"If a system of linear equations has at least 1 solution then the pivot can be parameterized in terms of non-pivot variables.
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That is, if xk1, . . . , xkr are the nonpivot variables of the
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system, then for any pivot variable xi, we can find constants aij ∈ R such that
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xpi = ai0 + ai1xk1 + · · · + airxkr"
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Theorem 1: Row Equivalence
If 2 matrixes are row equivalent, then the system of linear equations they represent(as augmented matrices) have the solution sets
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Theorem 2: Gauss-Jordan
"Every matrix A is row equivalent to a matrix B in row echelon form. Furthermore, if B is in RREF then B is unique
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* we call B the reduced row echelon form of A i.e B = rref(A)"
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Theorem 3: Rouché-Capelli
"Supposing a system has augmented matrix A and co-efficient matrix C
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1) The last column of rref(A) has pivot if and only if the system has no solutions
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2) The last column of rref(A) doesn’t have a pivot and every column of rref(C) has a pivot if and only if the system has exactly 1 solution
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3) The last column of rref(A) doesn’t have a pivot and there is a column of rref(C) without a solution if and only if the sytem has infinitely many solutions"