How to determine if a matrix has a unique solution. That is, it is of the form where is a subspace.
How to determine if a matrix has a unique solution. Be able to use rank of an augmented matrix to determine consistency or inconsistency of a system. Today’s Goals Be able to use rank of a matrix to determine if vectors are linearly independent. For each of these (augmented) matrices determine if the associated system is consistent, and if it is consistent, determine if the solution is unique a) infinitely many solutions because $x_1 = $ any number? Sep 17, 2022 · The first chapter of this text was spent finding solutions to systems of linear equations. This algebra video tutorial explains how to determine if a system of equations contain one solution, no solution, or infinitely many solutions. One may have wondered “Are the ideas of the first chapter related to what we have been doing recently?” The answer is yes, these ideas are related. 4. From the 1st row, x+2y+z = 7 ---- (1) From the 2nd row, 5y = 10 ---- (2) y = 2 From the 3rd row, -15z = -30 ---- (3) z = 2 By applying the value of y and z in (1), we get x + 2 (2) + 2 = 7 x + 6 = 7 x = 7 - 6 x = 1 x = 1, y = 2 Sep 29, 2022 · setup simultaneous linear equations in matrix form and vice-versa, understand the concept of the inverse of a matrix, know the difference between a consistent and inconsistent system of linear equations, and learn that a system of linear equations can have a unique solution, no solution or infinite solutions. Nov 3, 2019 · The question is based on an exercise in linear algebra: THOUGHTS: By looking at the determinant, I know that a unique solution occurs when $c \neq 2$ or $-3$ since a square matrix is singular i Apr 22, 2015 · The formula has det (A) in the denominator of the unique solution values, where A is the coefficient matrix (only the first 3 columns of your augmented matrix). The code should be such that you enter the coefficient matrix A and the corresponding column vector b. A homogeneous system always has $\vec 0$ as a particular solution, and the second theorem applies to homogeneous systems by taking $\vec p=\vec 0$. I mean, like in a homogeneous system of equations,if det (A)=0,then the system has infinite number of solutions else if det (A) is not zero then it has one a unique,but trivial solution. 1), but the left-hand-side of the third equation has been When a linear system has a unique solution, every column of the coefficient matrix has a pivot position. Dec 26, 2022 · 3. If the kernel of A contains only the null vector, i. Proof (2 3): The null space of a matrix is {0} if and only if the matrix has no free variables, which means that every column is a pivot column, which means A has n pivots. So, we have explained how to determine if a system of equations has the three types of solution which are a unique solution, no solution, or infinitely many solutions. Feb 2, 2021 · In this video, we use the determinant to ascertain whether a system of linear equations has a unique solution. We’d like to be able to determine if a solution exists; if so, determine if it is unique; compute a solution if one exists; find an orthonormal basis of the Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. 6. For a square matrix A, A system of m equations with n variables has a unique solution if and only if its augmented matrix has n pivot entries and no pivot entry in the rightmost column. A consistent system with no free variables has a unique solution. How to find it by matrix method ?#UniqueSo Rows of 0's kann be removed (or added if you so fancy) without changing the solutions of the system. The logic behind that it would have one solution is that you have Jun 9, 2021 · Write an M-file function, that tells whether a system of linear equations has no solution, unique solution, or infinitely many solutions. Uniqueness Theorem The uniqueness of a solution to a linear system of equations is an important concept in mathematics. Consider 1 + +, +, + Note that the first two equations are the same as in (2. The second condition implies that the matrix A is invertible. Jan 25, 2016 · If a matrix has a row of zeros but the number of variables is equal to the number of nonzero rows, does that mean the matrix has an infinite amount of solutions or a unique solution? Theorem 1. 5 Solution Sets of Linear Systems De nition. Learn how to find h and k such that a linear system of equations has no solution, one solution, or infinitely many solutions. 3. As a consequence, if n > m—i. In this video you will get to know that 1 How to check that given system of linear equation have unique solution?2. This solution might be a unique solution, where all of the variables have only one possible Nov 4, 2016 · The easiest way is to show that the corresponding coefficient matrix is non-singular. Put the associated augmented matrix in reduced row echelon form and find solutions, if any, in vector form. What conditions must be satisfied for the overdetermined system below to be consistent? Do not try to solve for the solution naively. Learning Objectives Interpret the row-reduced augmented matrix for a linear system to understand the system's solutions Identify basic and free variables for a linear system Determine whether a linear system is consistent or inconsistent Determine whether a linear system has no solutions, one unique solution, or infinitely many solutions The Story So Far In earlier lectures, we discussed Jul 28, 2023 · Every system of equations has either one solution, no solution, or infinitely many solutions. By definition, a system of linear equation is said to be "consistent" if and only if it has at least one solution; and it is "inconsistent" if and only if it has no solutions. X1 - X2 + X3 5X1 X2 + x3 = 6 4X1 3x2 + 3x3 = 0 The system has a unique solution because the determinant of the coefficient matrix is nonzero. Jun 5, 2016 · But then $\ d-nb = 0$, and the bottom row of the above matrix is all zeros. This video Linear Algebraic Equations For 2 Equations and 2 Unknowns, the solution is the intersection of the two lines. For example 1 2 —2 —3 in general, that Ax has a solution. Jul 23, 2025 · A system of linear equations will have exactly one solution under the following conditions: Unique Intersection: The graphs of the equations intersect at exactly one point. Sep 17, 2022 · The constants and coefficients of a matrix work together to determine whether a given system of linear equations has one, infinite, or no solution. Oct 22, 2021 · What is a systematic way of determining whether a system of equations has a unique solution? For example, this system of $2$ equations with $3$ unknowns: $$2a+b+c=0\\\\ 6a+2b+c=0. If the system is consistent does a unique Such a system has infinitely many solutions. I know this: A system has a unique solution when it is consistent and the number of variables is equal to the number of nonzero rows, while a system has infinitely many solutions when it is consistent and the number of variables is more than the number of nonzero rows and a system has no solution if the equations are inconsistent. 11 Solving RREF systems Suppose we start with a linear system with matrix form A 𝐱 = 𝐛 then put the augmented matrix (A ∣ 𝐛) into RREF. We can also determine whether a system has a unique solution or infinite number of solutions or no solution using the matrix equation. 5 Must Know Facts For Your Next Test For a linear system to have a unique solution, the equations must represent lines that intersect at exactly one point in a two-dimensional space. We will illustrate what happens during Gaussian elimination in these two cases. 1 Let x∈RN be a solution to the linear system Ax = b, where b∈RN, A is an N x N real matrix. This video walks through the process of using matrices and row Since a homogeneous system always has a solution (the trivial solution), it can never be inconsistent. Jun 16, 2016 · If we have 3 systems of equations, that intersect at the point (1,2,3) does it have trivial or nontrivial solutions? Let's assume it is this system of equations which intersects at (1,2,3) and row System of Equations (CHAPTER 5) Topic If a solution exists, how do we know if it is unique? Description Learn how we know if an existing solution is unique. 2, it will be shown why a solution set for a linear system contains either no solution, one solution, or infinitely many solutions. , it has the same number of rows and columns). Now, if we know the Recipe: Parametric Vector Form (Homogeneous Case) Let A be an m × n matrix. A matrix equation A X = B has a unique solution if the following conditions are met: The coefficient matrix A is square (i. Here we will prove that the resulting matrix is unique; in other words, the resulting matrix in reduced row-echelon does not depend upon the particular sequence of elementary So, in this way we have calculate infinite solutions of a matrix Note: To know about the infinite solution of a matrix first we have to check nonzero rows in the matrix. It involves the use of matrices and their ranks. Cramer's Rule provides a method to find the unique solution of a system of linear equations by using determinants when the determinant of the coefficient matrix is non-zero. Also, we can find the number of solutions by the graphical method. If consistent, is the solution Jul 23, 2025 · Determining whether a homogeneous linear system possesses a unique solution (trivial) or an infinite number of solutions (nontrivial) involves examining the determinant of the coefficient matrix. T is one-to-one. We use various methods, like graphing, substitution, elimination, or matrix operations, to find the nature or the actual solutions of the system. A linear system has either no solution or infinite number of solutions if and only if the matrix is singular. Aug 19, 2025 · (a) is an existence theorem. 3 Can someone please tell me what a matrix looks like when there is infinite solutions, unique solution and no solutions. For a square matrix A, The problem is that this book starts defining determinants, rather than discovering them as a way to discover if the system ahs an unique solution. (and for the records, a consistent system with $1$ or more free variables has infinitely many solutions. This video teaches you how we know if an existing solution is unique. Aug 30, 2023 · The solution of the system of linear equations will be unique if and only if the determinant of the coefficient matrix is nonzero. Homogeneous and inhomogeneous systems Jan 14, 2019 · An augmented matrix has an unique solution when the equations are all consistent and the number of variables is equal to the number of rows. The system is consistent and it has unique solution. This video explains how to determine if a system of equations has one solution, no solution, or infinite solutions from the an augmented matrix in RREF form. Define A = $\begin {bmatrix}1&2&1\\2&2&-2\\-1&2&-3\end {bmatrix}$ It is a property that if this matrix is non-singular, that the system of linear equations corresponding to this matrix has exactly one solution for any combination of outcomes. The equation $Ax=b$ can be written $f (x)=b$ and saying that this equation has a unique solution is only saying that $b$ has a unique preimage by $f$. Sep 20, 2016 · Any answers would be appreciated. Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. In this problem, we determine values of unknown constant k, if any, will give one solution (a unique solution), no solution infinitely, many solutions to the system of equations. The constants and coefficients of a matrix work together to determine whether a given system of linear equations has one, infinitely many, or no solution. As @Lazar commented, if there are additional conditions, you can get a unique solution. So in order to say that $f$ is bijective we need to have that for each $b$ the equation $f (x)=b$ has a unique solution $x$. Find the reduced row echelon form of A. The augmented matrix includes the coefficients of the variables on the left side of the equations and the constants on the right side of the equations 1 If we have a unique solution to Ax = b, then rref(A) is the matrix which has a leading 1 in every column. The process we use to find the solutions for a homogeneous system of equations is the same process we used in the previous section. , if the number of unknowns is larger than the number of equations—, then the system will have infinitely many solutions. Mar 31, 2016 · If you put a square matrix into reduced row-echelon form and it has rank $n$ then the reduced row-echelon form of that matrix is $\mathbb {I}_n$ the identity matrix, which is equivalent to having a unique solution to the system. But two other possibilities exist: there could be no solution, or an infinite number of solutions. If this determinant is zero, then the system has an infinite number of solutions. Note that for a given system, the vectors $\vec p$ and $\vec v_i$ are not unique. A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an m n matrix and 0 is the zero vector in Rm. Here's how it works: Step 1: Write the system of linear equations in augmented matrix form. The set of solutions to the above equation, if it is not empty, is an affine subspace. Therefore, in the case of systems of linear equations, one way to determine whether the system will have a unique solution is to draw simple graphs of the equations and see if the lines intersect at a single point. Sep 6, 2019 · Consider the following augmented matrices in which * denotes an arbitrary number and ️ denotes a nonzero number. Also, we can find whether the system of equations has no solution or infinitely many solutions by graphical method. Thanks in advance. For example, for these matrices: In general, we have existence of a solution when the data vector is in the column space of the target matrix $\mathbf {A}$. Probably the most straightforward method (to fully distinguish between the various possibilities) that I've seen is transforming the corresponding augmented matrix into row-reduced echelon form. While considering the system of linear equations, we can find the number of solutions by comparing the coefficients of the equations. 2 If det (A) = 0 there may be no solutions to the equation, or there may be an in ̄nite number of them. A x = b has a unique solution for each b in R n. If we know the how the rank of the matrix is related to a and b, we can determine (maybe not exactly) the number of solutions for the system. Feb 21, 2025 · Given a linear system $A x = b$ (where $A$ is an $n\times n$ matrix), if the system has a solution, then it is easy to decide whether this solution is unique: if $rank (A)=n$ then the solution is unique; if $rank (A)<n$ then there are infinitely many solutions. But in $6 \times 3$ matrix there can be $6$ unknowns, so is it possible that system can be unique, if yes how?. Therefore, a unique solution exists only if $\ ad - bc$ $\ne$ $\ 0$. In matrix form, a unique solution exists when the coefficient matrix has full rank, meaning its rank equals the number of variables. However, if the matrix is singular, there are two possibilities: either the system is consistent and has infinitely many solutions, or it is inconsistent and has no solutions Jun 2, 2019 · As per my understanding if rank of matrix $=$ no. You have at most one soulution, if the rank is equal to the number of columns (which is the number of variables). of unknowns, than system has unique solution. A nxn homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. Let us learn how to solve matrix equations in different methods along with examples. Edit: Yes, this particular system is unsolvable (thanks to Jack D'Aurizio and others), but I kind of wanted to know how to find a general way to calculate solutions or the absence of solutions when the determinant is zero and the matrix has no inverse. There can be many different sequences of row operatation that could be used to put the system into echelon form. Jun 22, 2025 · How can you determine if a linear system has infinitely many solutions directly from its reduced row echelon matrix? What can you say the solution space of a linear system if there are more unknowns than equations and at least one solution exists? Feb 17, 2020 · A linear system has a unique solution if and only if the matrix is non-singular. Associated with every square matrix is a scalar that is called the determinant of the matrix, and deter-minants have numerous conceptual and practical uses. Make a 18 Yes: by showing that the system is equivalent to one in which the equation $0=3$ must hold, you have shown the original system has no solutions. 2. Set of solutions Consider the linear equation in : where and are given, and is the variable. Simply put if the non-augmented matrix has a nonzero determinant, then it has a solution given by $\vec x = A^ {-1}\vec b$. Under certain conditions, ordinary di erential equation partial di erential equation and matrix equations will have unique solutions under the prescribed boundary condition and the driving source terms. Answers to the following two questions will determine the nature of the solution set for a linear system. 4. First, we construct the augmented matrix, given by [2 1 1 0 1 2 2 0] Then, we carry The third row of A is the sum of its first and second rows, so we know that if Ax = b the third component of b equals the sum of its first and second components. If b does not satisfy b3 = b1 + b2 the system has no solution. For matrices of this type it will not be the b1+ case, For the matrix A above if we take and get it into reduced row echelon form ( 1 0 1— b Learn how to analyze augmented matrices to determine if a linear system has no solution, exactly one solution, or infinitely many solutions. In this explainer, we will learn how to determine whether a linear system of equations has a unique solution, no solution, or an infinite number of solutions. We can find whether a homogeneous linear system has a unique solution (trivial) or an infinite number of solutions (nontrivial) by using the determinant of the coefficient matrix. The primary reason we are presenting the more general matrix case n ≥ 1 is apply to the standard second order scalar initial value problem What is a singular matrix and what does it represent?, What is a Singular Matrix and how to tell if a 2x2 Matrix or a 3x3 matrix is singular, when a matrix cannot be inverted and the reasons why it cannot be inverted, with video lessons, examples and step-by-step solutions. That is, if the data vector can be written as a linear combination of the fundamental columns of $\mathbf {A}$. If a combination of the rows of A gives the zero row, then the same combination of the entries of b must equal zero. 1: Solutions and the Reduced Row-Echelon Form of a Matrix As we have seen in earlier sections, we know that every matrix can be brought into reduced row-echelon form by a sequence of elementary row operations. , d e t (A) ≠ 0. In all other cases, it will have infinitely many solutions. Determine whether the given matrices are consistent. Since the linear system has a unique solution when $\operatorname {rank} A=n$, this can never occur. In the last section, we used the Gauss-Jordan method to solve systems that had exactly one solution. Graphically, a unique solution can be visually identified as the point where two An equation of the form ax + by + c = 0 where a, b, c ∈ R, a ≠ 0 and b ≠ 0 is a linear equation in two variables. Jun 20, 2024 · 1 4 1 By now, we have seen several examples illustrating how the reduced row echelon matrix leads to a convenient description of the solution space to a linear system. In this section, we will use this understanding to make some general observations about how certain features of the reduced row echelon matrix reflect the nature of the solution Sep 17, 2022 · Lemma 1. Otherwise, it is inconsistent ⇒ If the matrix corresponding to a set of linear equations is non-singular, then the system has one unique solution and is consistent. Use a calculator to find the determinant of an n × n matrix. We can see that the graphs of the equations of the system intersect at only one point. KA = 0, then this solution is unique (no other solutions exist). Determinant Method: In a system of linear equations represented in matrix form, the system has a unique solution if the determinant of the coefficient matrix is non-zero. The system can have a unique solution, infinite solutions, or no solution. Shared from Wolfram CloudIn Section 1. This is the manner of how we solve a boundary value problem. It guarantees that a solution exists on some open interval that contains x 0, but provides no information on how to find the solution, or to determine the open interval on which it exists. I have been searching the internet and I cannot find a straightforward answer of what the matrix should look like. I want to know what happens for the case of non-homogeneous equations. Sep 17, 2022 · Solution Notice that this system has m = 2 equations and n = 3 variables, so n> m. If jAj = 0, then Ax = b usually has no solutions, but does have solutions for some b. In this case, you would start with: $$\left [\begin {array} {ccc|c}1 & 3 & -1 & -4\\4 & -1 & 2 & 3\\2 & -1 & -3 & 1\end {array}\right]$$ Subtracting $4$ times the first row from the second, and $2 Given n equations and n unknowns, one usually expects a unique solution. Matrix Equation A system of equations can be solved using matrices by writing it in the form of a matrix equation. Solutions to Consistent Systems A consistent system of linear equations can have either: Unique Solution: This occurs when the system has exactly one solution. When working with a system of linear equations, it is easy to assume that there is a solution. 1. It follows from the discussion in this section is that two linear simultaneous equations in two unknowns can have a unique solution, no solution or infinitely many solutions and this is true for every system of linear simultaneous equations with \ (m\) equations and \ (n\) unknowns. In this article, we will The rank method is a method used to determine the number of solutions to a system of linear equations. I need to know how to find it. Therefore by our previous discussion, we expect this system to have infinitely many solutions. Feb 15, 2020 · But is there a way of finding the solution by only using row reductions on the augmented coefficient matrix? I seem to get stuck in my own thought process. Apply row reduction. The determinant of the coefficient matrix A is not zero, i. Put equations for all of the x i in order. Dependent: The system has infinitely many solutions. Thus a homogeneous system of equations always either has a unique solution or an infinite number of solutions. ρ (A) = ρ ( [A|B]) = 3. That means if the number of variables is more than nonzero rows then that matrix has an infinite solution. 19. T is onto. This section begins to show that relationship. Inhomogeneous systems: Ax = b has the unique solution x = A 1b, if jAj 6= 0. e. Is my line of reasoning to prove the question correct? Thanks for the help! How to Check Consistency of Linear Equations Using MatricesNumber of non zero rows are 3. Using Gaussian elimination can help determine if a system has a unique solution by transforming the system into row echelon form and checking for leading ones in each row. Mar 30, 2015 · Using determinants to find a unique solution Ask Question Asked 10 years, 6 months ago Modified 9 years, 9 months ago Use a calculator to find the determinant of an n × n matrix. T is invertible. (e) Use the determinant to determine whether a system of equations has a unique solution. The concept will be fleshed out more in later chapters, but in short, the coefficients determine whether a matrix will have exactly one solution or not. The statements (20) and (21) are proved, since we have a formula for the solution, and it is easy to see by multiplying A x = b by A 1 that if x is a solution, it must be of the form A homogeneous system of equations Ax = 0 will have a unique solution, the trivial solution x = 0, if and only if rank[A] = n. Without explicitly solving the system and based on the RREF of the augmented matrix of a system, determine if the system has no solution (inconsistent), unique solution (independent) or infinitely many solutions (dependent). ) 1. We have spent the first two sections of this chapter learning operations that can be performed with matrices. The entries of a matrix are denoted by aij, where i is the row number and j is the column number. In this section, we will determine the systems that have no solution, and solve the systems that have infinitely many solutions. Suppose that the free variables in the homogeneous equation A x → = 0 → are, for example, x 3, x 6, and x 8. This matrix is called the identity matrix. Suppose the resulting matrix in RREF is (A ′ ∣ 𝐛 ′). A matrix with one column is also called a column vector. When we consider a system of linear equations, we can find the number of solutions by comparing the coefficients of the variables of the equations. Write the parametric form of the solution set, including the redundant equations x 3 = x 3, x 6 = x 6, x 8 = x 8. That is, it is of the form where is a subspace. For 3 Equations and 3 Unknowns, the solution is the intersection of three planes. But Determine whether the given system has a unique solution, no solution, or infinitely many solutions. Apr 21, 2018 · To be more specific, I'm not having trouble with understanding solution types or how to solve an augmented matrix. The book calls such a matrix “tall and thin”. X1 X2 +X34 5x1 x2X3 6 4x1 5x2 5x3O0 e system has a unique solution because the determinant of the coeffidient matrix is nonzero. $$ @ ¡ @ @ ¡ ¡ @ ¡ ¡¡ @ ¡ ¡¡ ¡ @¡¡ ¡ ¡ ¡ @¡ No solutions Whether or not there is a unique solution depends on the determinant of A: 2 det (A) 6= 0 means there is a unique solution to A x = b. In this article, we will learn how to determine the number of solutions in a system of equations with two variables. But am I missing any scenarios? Are there other times when a So if you find out that the rank of the coefficient matrix is the same as the number of unknowns then it has a unique solution and if we find that the rank of A-that is the coefficient matrix- is less than the number of unknowns then we have infinite number of solutions. Sep 2, 2018 · I am reading this section of my text and I'm a bit confused as to why we're finding the determinant of the system to see if the solution is unique: Why does a nonzero determinant imply a unique sol The (unique) solution of this happens to be u1(t) = cos t, u2(t) = sin t, but the point of these notes is to consider equations where there may not be simple formulas for the solution. 1 may have. In other words, mainly questions two and three now. Thanks. The coefficient matrix is given by: Oct 13, 2013 · I'm supposed to determine if system Ax = b (where x and b have appropriate number of components) has a solution for all choices of b. As you can see, the final row of the row reduced matrix consists of 0. Since every row contains at most one pivot position, there must be at least as many rows as columns in the coefficient matrix. There is no unique solution to the above system of equations which means that my assumption that $\ ad - bc = 0$ is wrong. Jul 23, 2025 · Independent: The system has exactly one unique solution. See Recipe: Compute a Spanning Set for a Null Space in Section 2. Suppose we’re given an equa ion of the ere A is tall and thin. My question is that is it possible to determine the TYPE of solution (unique, consistent, no solution, etc) through determining the rank of the augmented matrix and comparing it to the rank of the matrix on its own. Moreover, (a) provides no information on the number of solutions that Equation 1. This means that for any value of Z, there will be a unique solution of x and y, therefore this system of linear equations has infinite solutions. 9rf3nca9w94nrfhak2euekiutb7iloyl528mvsl4aqv7jp3