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Chapter 3: Problem 10

Prove or give a counterexample to the following statement: If the coefficient matrix of a system of \(m\) linear equations in \(n\) unknowns has rank \(m\), then the system has a solution.

Short Answer

Expert verified
The given statement can be proved using the Rouch茅-Capelli theorem. The theorem states that a system of linear equations has a solution if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. Given that the rank of the coefficient matrix is $m$, we deduce that the rank of the augmented matrix must also be $m$ since it cannot be less than the rank of the coefficient matrix or more than $m+1$. Thus, according to the Rouch茅-Capelli theorem, the system indeed has a solution.

Step by step solution

01

Understanding the Rank

In linear algebra, the rank of a matrix is defined as the maximum number of linearly independent columns (or equivalently, rows) of the matrix. If the rank of the coefficient matrix is equal to m, it means that there are m linearly independent rows in the matrix.
02

Understanding the Rouch茅鈥揅apelli theorem

The Rouch茅鈥揅apelli theorem is a fundamental theorem in linear algebra that provides a criterion for the existence and uniqueness of solutions of a system of linear equations. The theorem states that a system of linear equations is consistent (having a solution) if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. Moreover, if the rank equals the number of unknowns (n), then the system has a unique solution.
03

Proving the given statement using the Rouch茅鈥揅apelli theorem

According to the Rouch茅鈥揅apelli theorem, a system of linear equations has a solution if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. Given that the rank of the coefficient matrix is m in the exercise's statement, we need to show that the rank of the augmented matrix is also m. We know that the rank of the augmented matrix cannot be less than the rank of the coefficient matrix, since the coefficient matrix is contained within the augmented matrix. Thus, the rank of the augmented matrix must be either m or m+1. However, if the rank of the augmented matrix is m+1, then it would be inconsistent, as there would be a row with the last element being a nonzero number while all the corresponding coefficients are zero. So, the rank of the augmented matrix can only be m in this case. This implies that the system of linear equations has a solution, according to the Rouch茅鈥揅apelli theorem. Therefore, we have proved that if the coefficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution.

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Most popular questions from this chapter

For each of the following matrices, compute the rank and the inverse if it exists. (a) \(\left(\begin{array}{ll}1 & 2 \\ 1 & 1\end{array}\right)\) (b) \(\left(\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right)\) (c) $\left(\begin{array}{rrr}1 & 2 & 1 \\ 1 & 3 & 4 \\ 2 & 3 & -1\end{array}\right)$ (d) $\left(\begin{array}{rrr}0 & -2 & 4 \\ 1 & 1 & -1 \\ 2 & 4 & -5\end{array}\right)$ (e) $\left(\begin{array}{rrr}1 & 2 & 1 \\ -1 & 1 & 2 \\ 1 & 0 & 1\end{array}\right)$ (f) $\left(\begin{array}{lll}1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1\end{array}\right)$ (g) $\left(\begin{array}{rrrr}1 & 2 & 1 & 0 \\ 2 & 5 & 5 & 1 \\ -2 & -3 & 0 & 3 \\ 3 & 4 & -2 & -3\end{array}\right)$ (h) $\left(\begin{array}{rrrr}1 & 0 & 1 & 1 \\ 1 & 1 & -1 & 2 \\ 2 & 0 & 1 & 0 \\ 0 & -1 & 1 & -3\end{array}\right)$

Let \(A\) be an \(m \times n\) matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms \(A\) into an upper triangular matrix.

Let $$ A=\left(\begin{array}{rrr} 1 & 2 & 3 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{array}\right), B=\left(\begin{array}{rrr} 1 & 0 & 3 \\ 1 & -2 & 1 \\ 1 & -3 & 1 \end{array}\right), \text { and } C=\left(\begin{array}{rrr} 1 & 0 & 3 \\ 0 & -2 & -2 \\ 1 & -3 & 1 \end{array}\right) \text {. } $$ Find an elementary operation that transforms \(A\) into \(B\) and an elementary operation that transforms \(B\) into \(C\). By means of several additional operations, transform \(C\) into \(I_{3}\).

Suppose \(u\) and \(v\) are distinct vectors. Show that, for distinct scalars \(k\), the vectors \(u+k(u-v)\) are distinct.

Prove Theorem 3.15. Let \(v_{0}\) be a particular solution of \(A X=B\), and let \(W\) be the general solution of \(A X=0 .\) Then \(U=v_{0}+W=\left\\{v_{0}+w: w \in W\right\\}\) is the general solution of \(A X=B.\)
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