 | Maxime Bôcher, Edmund Pendleton Randolph Duval - Algebra - 1907 - 344 pages
...one solution. Hence the given system of equations is consistent, and we have the theorem: THEOREM 1. A necessary and sufficient condition for a system...to be consistent is that the matrix of the system have the same rank as the augmented matrix, From the foregoing considerations \ve have also THEOREM... | |
 | Jacob William Albert Young - Mathematics - 1911 - 434 pages
...does not vanish. In this way we obtain one and only one value for each of the unknowns zi, . . . , xr. The preceding considerations prove the following...the augmented matrix. Since the values assigned to zr+i . . . xn arc arbitrary, it also follows that a system of linear equations has an infinite number... | |
 | Arthur Dodd Snyder - 1919 - 110 pages
...gives values for г of tha x's as soon as we assign values to the (n - г) х|в. Hence: THEOREM jß. A necessary and sufficient condition for a system...to be consistent is that the matrix of the system have the same rank as the augmented matrix. _2. If in a system of linear equations the matrix of the... | |
 | Leonid Mirsky - Mathematics - 1990 - 468 pages
...OF COEFFICIENTS, while a . . . a1n 61 is known as the AUGMENTED MATRIX. Theorem 5.3.1. (Consistency theorem) A necessary and sufficient condition for...linear equations to be consistent is that the matrix of coefficients should have the same rank as the augmented matrix. The system of equations (5.3.1) is,... | |
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In this way we obtain one and only one value for each of the unknowns x\, . . . , xr. The preceding considerations prove the following theorem: A necessary and sufficient condition for a system of linear equations to be consistent is that the matrix of... 