

What is a Matrix? An Introduction
An arrangement of numbers, expressions, or symbols in a rectangular array is called a matrix (plural: matrices). With an order of the number of rows × the number of columns, this layout comprises horizontal and vertical columns. In three-dimensional space, every pair of points represents a different equation with one or more solutions.
Matrix algebra involves matrix operations like addition, subtraction, multiplication, etc.
Inverse of Matrix
There exists an inverse matrix A-1 if A is a non-singular square matrix, which satisfies the below condition:
AA-1 = A-1A = I, where I’ is an Identity matrix.
Elementary operations
A matrix is subjected to six operations or transformations, three caused by the rows and three by the columns. These actions are referred to as elementary actions. Only square matrices are used for these operations.
These elementary operations are:
Interchanging any two rows or columns ($R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_j$)
Multiplying the elements of any row or column by a positive integer ($R_i \rightarrow kR_i$ or $C_i \rightarrow kC_i$)
Addition or subtraction of multiples of one row to another ($R_i \rightarrow R_i + kR_j$ or $C_i$ → $C_i+ kC_j$)
How to find the inverse of the 3×3 matrix?
In order to determine a matrix's inverse, we must do the following steps:
Finding the minors matrix is the first step.
Change that matrix into a matrix of cofactors
Find the adjoint of the matrix
Multiply by \[\dfrac{1}{\text{determinant}}\]
Finding Inverse of Matrix Using Elementary Operations
If the three matrices, X, A, and B, are all of the same order, then X = AB is an equation of matrices. The basic row operations will be applied simultaneously to the matrices X and A, which is on the RHS side of the matrix equation of the product of AB for the given matrix equation.
Similarly, we can apply the basic column operations simultaneously on the matrices X and B (on the RHS side of the matrix equation) of the product of AB.
Thus, in general, if we calculate the inverse of a matrix A by doing simple row operations on A = IA in a certain order until we get: I = BA.
Also, the value of the inverse of matrix A must be obtained by performing basic column operations on the value of A = AI sequentially until we reach I = AB.
Inverse of Matrix Using Elementary Row transformation
As the names suggest, only the rows of the matrices are changed; the columns remain unchanged. A certain set of guidelines is followed while performing these row operations to ensure that the converted matrix is identical to the original matrix. Let’s understand how to find the inverse of a matrix using elementary row operations with the help of an example:
Example:
A = \[\left[ {\begin{array}{*{20}{c}}0&1&2\\1&2&3\\3&1&1\end{array}} \right]\]
Applying $R_1 \rightarrow R_2$
\[\left[ {\begin{array}{*{20}{c}}1&2&3\\0&1&2\\3&1&1\end{array}} \right]\]
Inverse of Matrix Using Elementary Column Transformation
The basic column operation is obtained by similarly applying those three-row operations to columns.
Solved Examples:
1. Find the inverse of the matrix using elementary operations.
A = \[\left[ {\begin{array}{*{20}{c}}0&1&2\\1&2&3\\3&1&1\end{array}} \right]\]
Solution:
A = \[\left[ {\begin{array}{*{20}{c}}0&1&2\\1&2&3\\3&1&1\end{array}} \right]\]
Using the elementary row operation, we know;
$A = IA$
\[\left[ {\begin{array}{*{20}{c}}0&1&2\\1&2&3\\3&1&1\end{array}} \right]\] = \[\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\]. A
Applying $R_1 \rightarrow R_2$
\[\left[ {\begin{array}{*{20}{c}}1&2&3\\0&1&2\\3&1&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0&1&0\\1&0&0\\0&0&1\end{array}} \right].A\]
Applying $R_3 \rightarrow R_3 - 3R_1$
\[\left[ {\begin{array}{*{20}{c}}1&2&3\\0&1&2\\0&{ - 5}&{ - 8}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0&1&0\\1&0&0\\0&{ - 3}&1\end{array}} \right].A\]
Applying $R_1 \rightarrow R_1-2R_2$ and $R_3\rightarrow R-3 + 5R_2$
$\begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 2 \end{bmatrix}= \begin{bmatrix} -2 & 1 & 0 \\ 1 & 0 & 0 \\ 5 & -3 & 1 \end{bmatrix} \cdot A$
Applying $ R_3 \rightarrow \frac{R_3}{2}$
\[\left[ {\begin{array}{*{20}{c}}1&0&{ - 1}\\0&1&2\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 2}&1&0\\1&0&0\\{\dfrac{5}{2}}&{\dfrac{{ - 3}}{2}}&{\dfrac{1}{2}}\end{array}} \right].A\]
Applying $R_1 \rightarrow R_1+R_3, R_2 \rightarrow R_2 – 2R_3$
\[\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\dfrac{1}{2}}&{\dfrac{{ - 1}}{2}}&{\dfrac{1}{2}}\\{ - 4}&3&1\\{\dfrac{5}{2}}&{\dfrac{{ - 3}}{2}}&{\dfrac{1}{2}}\end{array}} \right].A\]
Therefore,
$A^{-1}=\begin{bmatrix}{\dfrac{1}{2}}&{\dfrac{{ - 1}}{2}}&{\dfrac{1}{2}}\\{ - 4}&3&1\\{\dfrac{5}{2}}&{\dfrac{{ - 3}}{2}}&{\dfrac{1}{2}}\end{bmatrix}$
2. Find the inverse of the following matrix by elementary operations?
$\begin{bmatrix} -2 & 1 & 3 \\ 0 & -1 & 1 \\ 1 & 2 & 0 \end{bmatrix}$
Solution: First rewrite the matrix as:
$[A|I]=\left[\begin{array}{ccc:ccc} 1 & 0 & 0 & -2 & 1 & 3 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 0 & 0 & 1 & 1 & 2 & 0 \end{array}\right]$
Applying $R_3 \rightarrow 2R_3+R_1$
$\left[\begin{array}{ccc:ccc} 1 & 0 & 0 & -2 & 1 & 3 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 1 & 0 & 2 & 0 & 5 & 3 \end{array}\right]$
Applying $R_1$ → $R_1 + R_2$ and $R_3$ → $R_3+ 5R_2$
$\left[\begin{array}{ccc:ccc} 1 & 1 & 0 & -2 & 0 & 4 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 1 & 5 & 2 & 0 & 0 & 8 \end{array}\right]$
Applying $R_1$ → $2R_1 - R_3$ and $R_2$ → $8R_2 - R_3$
$\left[\begin{array}{ccc:ccc} 1 & -3 & -2 & -4 & 0 & 0 \\ -1 & 3 & -2 & 0 & -8 & 0 \\ 1 & 5 & 2 & 0 & 0 & 8 \end{array}\right]$
Applying $R_1$ → $\frac{1}{-4}, R_2$ → $\frac{1}{-8}$, and $R_3$ → $\frac{1}{8} $
$\left[\begin{array}{ccc:ccc} -1 / 4 & +3 / 4 & +2 / 4 & 1 &0 &0 \\ +1 / 8 & -3 / 8 & 2 / 8 & 0 &1 & 0 \\ 1 / 8 & 5 / 8 & 2 / 8 & 0 & 1 & 0 \end{array}\right]$
Hence $A^{-1}=\left[\begin{array}{ccc:ccc} -1 / 4 & 3 / 4 & 2 / 4 \\ 1 / 8 & -3 / 8 & 2 / 8 \\ 1 / 8 & 5 / 8 & 2 / 8 \end{array}\right]$
Important formulas
$AA^{-1} = A^{-1} A = I$, where I’ is an Identity matrix.
$A^{-1} = \dfrac{(adjA)}{(\det A)}$
Interesting facts
If A and B are square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A-1 and A is the inverse of B.
The inverse of the inverse matrix is equal to the original matrix.
If A and B are invertible matrices, then AB is also invertible.
Summary
We learn about the Inverse of a matrix and how to find the inverse of a matrix using elementary operations. An arrangement of numbers, expressions, or symbols in a rectangular array is called a matrix and there exists an inverse matrix $A^{-1}$ if A is a non-singular square matrix, which satisfies the $AA^{-1} = A^{-1} A = I$ condition. A matrix is subjected to six elementary operations.
Problems for solving
Find the inverse of matrix A using elementary column operation.
A =\[\left[ {\begin{array}{*{20}{c}}3&5&1\\4&3&1\\9&8&1\end{array}} \right]\]
Find the inverse of matrix A using elementary column operation.
A = \[\left[ {\begin{array}{*{20}{c}}1&0&0\\0&0&1\\0&1&1\end{array}} \right]\]
Answer:
\[\left[{\begin{array}{*{20}{c}}{-\frac{1}{3}}&{\frac{1}{5}}&{\frac{2}{{15}}}\\{\frac{1}{3}}&{ - \frac{2}{5}}&{\frac{1}{{15}}}\\{\frac{1}{3}}&{\frac{7}{5}}&{ - \frac{{11}}{{15}}}\end{array}} \right]\]
\[\left[ {\begin{array}{*{20}{c}}1&0&0\\0&{ - 1}&1\\0&1&0\end{array}} \right]\]
FAQs on Inverse of Matrix by Elementary Operations
1. What are the elementary operations or transformations of a matrix?
There are three main types of elementary operations that can be applied to the rows (or columns) of a matrix. These are:
- Interchanging any two rows or any two columns.
- Multiplying all the elements of any row or column by a non-zero number.
- Adding the elements of one row or column to the corresponding elements of another row or column after multiplying them by a non-zero number.
2. How do you find the inverse of a 3x3 matrix using elementary row operations?
To find the inverse of a 3x3 matrix, let's call it A, you start by writing the equation A = IA, where I is the 3x3 identity matrix. The goal is to apply a series of elementary row operations to the matrix A on the left side until it becomes the identity matrix (I). Crucially, you must apply the exact same operations in the same order to the matrix I on the right side. When the left side becomes I, the matrix on the right side will be the inverse, A⁻¹.
3. Can you give a simple example of finding an inverse for a 2x2 matrix?
Certainly. Imagine we have a 2x2 matrix A. We set up the equation A = I A, which looks like the original matrix on the left and the identity matrix on the right. Our aim is to turn the left matrix into the identity matrix [[1, 0], [0, 1]] using row operations. For instance, we might divide the first row to make the top-left element '1'. We must then perform the same division on the first row of the matrix on the right. After a few steps, when the left side is the identity matrix, the transformed matrix on the right is the inverse of A.
4. Why is the elementary operations method important for finding a matrix inverse?
This method is fundamental because it provides a systematic, step-by-step algorithm to find an inverse. It's not just a formula. This approach is highly valuable because it forms the basis for many computer algorithms used to solve complex systems of linear equations in fields like engineering, computer science, and economics. It provides a clear procedure that works for any invertible square matrix.
5. How do elementary row operations affect the determinant of a matrix?
The effect on the determinant depends entirely on the operation used:
- Interchanging two rows: The sign of the determinant is reversed (it is multiplied by -1).
- Multiplying a row by a non-zero number 'k': The new determinant becomes 'k' times the original determinant.
- Adding a multiple of one row to another row: The determinant remains unchanged. This is a very useful property in calculations.
6. Can you use both row and column operations at the same time to find an inverse?
No, this is a very important rule. You must choose one method and stick with it. If you begin finding the inverse using elementary row operations, you must continue using only row operations for the entire process. If you start with elementary column operations, you must use only column operations. Mixing them will result in an incorrect answer.
7. How can you tell if a matrix inverse doesn't exist while using this method?
You can identify that a matrix has no inverse if, at any point during the application of elementary row operations, you obtain a row consisting of all zeros on the left-hand side matrix. If this happens, you should stop the process. It signifies that the original matrix is a singular matrix (its determinant is zero) and therefore, an inverse does not exist.

















