Matrix Equation

A system of equations can be solved using matrices by writing it in the form of a matrix equation. We can also determine whether a system has a unique solution or infinite number of solutions or no solution using the matrix equation.

Let us learn how to solve matrix equations in different methods along with examples.

A matrix equation is of the form AX = B where A represents the coefficient matrix, X represents the column matrix of variables, and B represents the column matrix of the constants that are on the right side of the equations in a system. Let us consider a system of n nonhomogenous equations in n variables.

a₁₁ x₁ + a₁₂ x₂ + ... + a₁ₙ xₙ = b₁

a₂₁ x₁ + a₂₂ x₂ + ... + a₂ₙ xₙ = b₂

....

aₙ₁ x₁ + aₙ₂ x₂ + ... + aₙₙ xₙ = bₙ

Then the matrix equation that corresponds to the above system is:

AX = B, where

• A = A matrix made of coefficients = \(\left[\begin{array}{cccc}
  a_{11} & a_{12} & \cdots & a_{1 n} \\
  a_{21} & a_{22} & \cdots & a_{2 n} \\
  \vdots & \vdots & \ddots & \vdots \\
  a_{n 1} & a_{n 2} & \cdots & a_{n n}
  \end{array}\right]\)
• X = Column matrix of variables = \(\left[\begin{array}{c}
  x_{1} \\
  x_{2} \\
  \vdots \\
  x_{n}
  \end{array}\right]\)
• B = Column matrix of constants = \(\left[\begin{array}{c}
  b_{1} \\
  b_{2} \\
  \vdots \\
  b_{n}
  \end{array}\right]\)

To write a system of equations as a matrix equation:

• Make sure that the variables are in the same order in each of the equations.
• Make sure that only the variable terms are on the left side and the constant terms are on the right side.
• Then find the coefficient matrix A, variable matrix X, and the constant matrix B to write the given system as matrix equation AX = B.

Here is an example to understand these steps.

Example: Write the following system as a matrix equation.

3x + y + 2z - 10 = 0
z - 3y = -5
y + x - 2z = 7

Solution:

Let us get all variable terms on one side (also in order) and constants on the right side.

3x + y + 2z = 10
0x - 3y + z = -5
x + y - 2z = 7

Now we can write this system as:

\(\left[\begin{array}{ccc}
3 & 1 & 2 \\
0 & -3 & 1 \\
1 & 1 & -2
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{c}
10 \\
-5 \\
7
\end{array}\right]\)

This is the matrix equation of the given system which alternatively can be written as AX = B where, A = \(\left[\begin{array}{ccc}
3 & 1 & 2 \\
0 & -3 & 1 \\
1 & 1 & -2
\end{array}\right]\), X = \(\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]\), and B = \(\left[\begin{array}{c}
10 \\
-5 \\
7
\end{array}\right]\).

Let us solve the matrix equation AX = B for X. For this, we left multiply both sides of the equation by the inverse of A (that can be written as A^-1).

A^-1 (AX) = A^-1 B

We know that A^-1A = I, where I is the identity matrix of the same order as A.

IX = A^-1B

We also know that IX = X.

X = A^-1B

This gives the solution of the matrix equation. This is also known as inverse matrix equation and hence the process of using the above formula to solve a system of equations is known as the "inverse matrix method". Thus, here are the steps to solve a system of equations using matrices:

1. Write the system as matrix equation AX = B.
2. Find the inverse, A^-1.
3. Multiply it by the constant matrix B to get the solution. i.e., X = A^-1B.

We can see the examples of solving a system using these steps in the "Matrix Equation Examples" section below.

We know that we can find the inverse of a matrix only when it is nonsingular. i.e., A^-1 exists only when det (A) ≠ 0. Thus, the solution (X = A^-1B) exists and it is unique only when det (A) ≠ 0.

In case, if det (A) = 0, how can we know how many solutions a system may have? In this case, we have to find (adj A) B, where adj A stands for "adjoint of A".

• If (adj A) B ≠ O, then the system has no solution and hence the system is said to be inconsistent.
• If (adj A) B = O, then the system may be "consistent with an infinite number of solutions" or "inconsistent (no solution)".

Here, 'O' is a null matrix. This is summarized in the flowchart below.

☛ Related Topics:

• Homogeneous System of Linear Equations
• Matrix Calculator
• Matrix Formula
• Matrix Multiplication Calculator

What is the Definition of Matrix Equation?

A matrix equation is of the form AX = B and it is writing the system of equations as a single equation in terms of matrices. Here,

• A = A matrix formed by the coefficients
• X = A column matrix formed by the variables
• B = A column matrix formed by the constants

How to Solve Matrix Equation AX = B?

To solve a matrix equation AX = B:

• Find A^-1 (using the formula A^-1 = (adj A) / (det A).
• Find the solution using X = A^-1 B.

How to Solve System of Equations Using Matrix Equation?

To solve a system of equations using matrices:

• First, write all the variables on one side and the constants on the other side of the equations. Also, write the variables in the same order in every equation.
• Write the system in the form AX = B by writing all coefficients of variables in matrix A, all variables as a column in matrix X, and all constants in a column in matrix B.
• Find the solution using X = A^-1 B.

What are the Methods Used to Solve a Matrix Equation?

Here are the methods to solve a matrix equation AX = B.

• Matrix inversion method (using the formula X = A^-1 B)
• Cramer's rule. Click here to learn about it.
• Gauss Jordan method.

How to Write a System of Equations as a Matrix Equation?

We can write a system of equations as a matrix equation AX = B. Here are the steps for the same:

• Maintain the order of the variables to be the same in all the equations.
• Get all variables to the left side and send the constants to the right side of every equation.
• Write the matrix A with all the coefficients of variables, where each row of A represents the coefficients of variables in one equation.
• Write the matrix X with variables in order as a column.
• Write the matrix B with constants in order as a column.

How Many Solutions a Matrix Equation Has?

A matrix equation AX = B has:

• a unique solution (consistent) if det (A) ≠ 0.
• no solution (inconsistent) if det (A) = 0 and (adj A) B ≠ O.
• infinite solutions (consistent) or no solution (inconsistent) if det (A) = 0 and (adj A) B = O.

Here, O is the null matrix.

What is the Relation Between Matrix Equation and Rank of a Matrix?

For a matrix equation AX = B, [A B] represents the augmented matrix. Let r(A) and r(A, B) represent the ranks of matrices A and [A B].

• If r(A) = r(A, B) = number of variables, then the system is consistent and has unique solution.
• If r(A) = r(A, B) < number of variables, then the system is consistent and has infinitely many solutions.
• If r(A) ≠ r(A, B), then the system is inconsistent and has no solution.