The coefficient matrix of a linear system has four rows and four columns.

1) How may equations are there in this linear system?
2) How may variables are there in this linear system?
3) If this coefficient matrix has a pivot in each column, how many
solutions does the system have? (Does it have no solutions, one
solution, or many solutions?) Why?

Please help me ;)

I saw you posted that yesterday @Ice cream.

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1) The number of equations in a linear system is equal to the number of rows in the coefficient matrix. In this case, since the coefficient matrix has four rows, there are four equations in this linear system.

2) The number of variables in a linear system is equal to the number of columns in the coefficient matrix. In this case, since the coefficient matrix has four columns, there are four variables in this linear system.

3) If the coefficient matrix has a pivot in each column, it means that each column of the matrix has a leading entry (a non-zero entry in the row position of the pivot). When the coefficient matrix has a pivot in each column, it implies that there are no rows of zeros in the matrix.

In this case, since the coefficient matrix is a 4x4 matrix and it has a pivot in each column, it means that there are no rows of zeros. This condition implies that the system of equations is consistent, which means it has at least one solution.

If the coefficient matrix has the same number of columns as variables (four columns in this case), and each column has a pivot, then the system of equations has a unique solution. This is because each variable has a unique leading entry in each column, which leads to a unique solution for the system.

So, when the coefficient matrix has a pivot in each column, the system of equations has exactly one solution.

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#1. Really? You just said it had 4 rows, so ...

#2, Really, you just said it had 4 columns, so ...
#3. Looks like you better read up on what such a set of pivots means
google "coefficient matrix has a pivot in each column" for several discussions.