Let A be an m×n matrix and b ∈ R
m. Prove that the system of equations Ax = b is inconsistent
if and only if there is a leading term in the last column of row echelon form of the augmented matrix
To prove the statement, let's assume that the system of equations Ax = b is inconsistent. This means that there is no solution for the system of equations.
We know that the augmented matrix of the system can be written as [A | b].
When we row reduce the augmented matrix to its row echelon form, the leading terms (the leftmost non-zero entry in each row) will follow a certain pattern. If there is a leading term in the last column of the row echelon form, it means that there is a non-zero entry in the last column of the last row.
Now, since the system of equations is inconsistent, it means that there is a row in the row echelon form that corresponds to an equation of the form 0x1 + 0x2 + ... + 0xn = d, where d is a non-zero number. In other words, there is a row in the row echelon form that has all zero entries except for a non-zero entry in the last column.
Since the row echelon form represents an equivalent system of equations, this means that there is an equation in the system of equations of the form 0x1 + 0x2 + ... + 0xn = d, where d is a non-zero number. But this equation is contradictory since it implies that 0 = d, which is not possible.
Therefore, if there is a leading term in the last column of the row echelon form of the augmented matrix, the system of equations Ax = b is inconsistent.
Conversely, if the system of equations Ax = b is inconsistent, it means that there is no solution. This implies that in the row echelon form of the augmented matrix, there is a row of the form [0 0 ... 0 | d], where d is non-zero.
Since the last column of the row echelon form corresponds to the constants in the system of equations, a non-zero entry in the last column implies that there is a leading term in the last column.
Therefore, if there is a leading term in the last column of the row echelon form of the augmented matrix, the system of equations Ax = b is inconsistent.
Hence, we have shown that the system of equations Ax = b is inconsistent if and only if there is a leading term in the last column of the row echelon form of the augmented matrix.
To prove that the system of equations Ax = b is inconsistent if and only if there is a leading term in the last column of the row echelon form of the augmented matrix, we need to understand the concept of inconsistency and row echelon form.
First, let's define what it means for a system of equations to be inconsistent. A system of equations is inconsistent if there is no solution that satisfies all of the equations simultaneously. In other words, the system is impossible to solve.
Now, let's discuss row echelon form. Row echelon form is a way of representing a matrix where each row has zeros below the leading entry (the first nonzero entry) of the row above it. Additionally, the leading entry in each row is to the right of the leading entry in the row above it. This form simplifies solving systems of linear equations.
To prove the statement, we'll consider the augmented matrix [A | b] and perform row operations to transform it into row echelon form. If there is a leading term in the last column of the row echelon form, we can conclude that the system of equations is inconsistent.
First, assume that the system of equations Ax = b is inconsistent. This means that no solution satisfies all of the equations simultaneously. Thus, row operations on the augmented matrix can be performed until we obtain a row with all zeros except for a non-zero entry in the last column. This leading term indicates inconsistency since no value of x can satisfy this equation.
On the other hand, suppose that there is a leading term in the last column of the row echelon form of the augmented matrix. This implies that a row operation has been performed, leading to an equation of the form 0x + ... + 0x + 1 * c = d, where c and d are constants. This equation is inconsistent since it implies that 0x + ... + 0x + c = d, which cannot be satisfied for any value of x.
Therefore, we have shown that the system of equations Ax = b is inconsistent if and only if there is a leading term in the last column of the row echelon form of the augmented matrix.