A \"coefficient matrix\" refers to the matrix of values \"from the left side\" o

Question

A "coefficient matrix" refers to the matrix of values "from the left side" of the equations when written in standard form. For instance, for the system 1x_1 + 2x_2 = 3 4x_1 + 5x_2 = 6 the coefficient matrix is [1 4 2 5]. It is aptly named the coefficient matrix because it is the matrix of only coefficients of variables, and does not include the standalone constant terms. Suppose a system of equations has a 4 times 6 size coefficient matrix with four pivot positions. Is the system consistent? Why or why not?

Explanation / Answer

Lets suppose the coefficient matrix is m×n and think about the m×(n+1) augmented matrix. If you take row operations that reduce the coefficient matrix and apply them to the augmented matrix you will get an m × (n + 1) matrix lets it would be A whose leftmost n columns are an echelon form of the coefficient matrix. Because the coefficient matrix has a pivot in every row every row of A is nonzero and the leftmost nonzero entry of each row of A is in one of the leftmost n columns. That means A is in echelon form and has no pivots in the rightmost column, so the system is consistent.

for taking account the above question.....................

Lets take the 4×7 augmented matrix. If you take row operations that reduce the coefficient matrix and apply them to the augmented matrix you will get a 4×7 matrix that let it be A whose leftmost 6 columns are an echelon form of the coefficient matrix. Each pivot column of the coefficient matrix has only one pivot in it so the coefficient matrix has 4 pivots. Each of these pivots have to be in a different row so each of the 4 rows of coefficient matrix have a pivot. That means that all 4 rows of A are nonzero and that the leftmost nonzero entry of each row of A is in one of the leftmost 6 columns. That means A is in echelon form and has no pivots in the rightmost column, so the system is consistent .

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