Introduction
Solving the Inverse of 3 by 3 Matrix may seem tough for some students and hence they might need some Inverse of 3 by 3 Matrix – Solved Examples that will help them understand the concepts and solve the questions well. And for the same reason, we have provided the Inverse of 3 by 3 Matrix sample examples along with some great exercises that you can solve by yourself via Vedantu which will not only help you understand the concepts well but will also prepare you for the exams. Inverse operations are something that is commonly used in algebra to simply what might otherwise be a very complicated question.
Tips to solve the Inverse of 3 by 3 Matrix
Even though it may seem hard, the best method to overcome it is by solving the question over and over again by using a sample problem.
Write down all the steps and check the answer once again after solving it on your own.
The inverse of a 3 by 3 matrix is a bit complicated task but can be estimated by following the steps given below. A 3 by 3 matrix includes 3 rows and 3 columns. Elements of the matrix are the numbers that form the matrix. A single matrix is one whose determinant is not equivalent to zero. For each x x x square matrix, there exists an inverse of each matrix. The inverse of matrix x * x is represented by X. The inverse of a matrix cannot be easily calculated using a calculator and shortcut method.
\[ XX^{-1} = X^{-1} X = I_{2}\]
In the above property \[I_{2}\] indicates x * x matrix. For example, let us take 2 * 2 matrix as
\[\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}\]
Any x * x square matrix X, which has zero determinant always includes an inverse \[X^{-1}\].
It is appropriate for almost all the square matrices and is given by \[ XX^{-1} = X^{-1} X = I_{2}\]
How do you Find the Inverse of the 3 by 3 Matrix?
Here, you can see the inverse of 3 by 3 matrix steps to find the inverse of 3 by 3 matrix online
Estimate the determinant of the given matrix
Find the transpose of the given matrix
Calculate the determinant of the 2 x 2 matrix.
Prepare the matrix of cofactors
At the last, divide each term of the adjugate matrix by the determinant
Inverse Matrix Formula
The first step is to calculate the determinant of the 3 * 3 matrix and then find its cofactors, minors, and adjoint and then include the results in the below-given inverse matrix formula.
\[A^{-1} = 1/ |A | Adj (A)\]
Inverse of 3 X 3 Matrix Example
Let us solve the 3 X 3 matrix
\[\begin{bmatrix} a & b &c \\ d & e & f\\ g & h & i\end{bmatrix}\]
Examine the given 3 X 3 matrix
A = \[\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4\\ 5 & 6 & 0\end{bmatrix}\]
Let’s learn the steps to find the inverse of 3 X 3 matrices online
Examine whether the given matrix is invertible
This can be proved if its determinant is non-zero. If the determinant of the given matrix is zero, then there will be no inverse of the given matrix.
det(A) = 1(0-24) – 2(0-20) + 3(0-5)
det(A) = -24 + 40 – 15
det (A) = 1
As the value determinant is 1, we can say that a given matrix has an inverse matrix.
Finding the Transpose of the Given Matrix
To find the transpose of the given 3 X 3 matrix
Hence,\[A^{-1}\] = \[\begin{bmatrix} 1 & 0 & 5 \\ 2 & 1 & 6\\ 3 & 4 & 0\end{bmatrix}\]
Determining the determinants of the 2 X 2 minor matrices.
Now, we will determine the determinant of each and every 2 X 2 minor matrices.
For first row elements
\[\begin{bmatrix} 1 & 6\\ 4 & 0\end{bmatrix}\] = -24
\[\begin{bmatrix} 12 & 6\\ 3 & 0\end{bmatrix}\] = -18
\[\begin{bmatrix} 2 & 1\\ 3 & 4\end{bmatrix}\] = 5
For second-row elements
\[\begin{bmatrix} 0 & 5\\ 4 & 0\end{bmatrix}\] = -20
\[\begin{bmatrix} 1 & 5\\ 3 & 0\end{bmatrix}\] = -15
\[\begin{bmatrix} 1 & 0\\ 3 & 4\end{bmatrix}\] = 4
For third-row element
\[\begin{bmatrix} 10 & 65\\ 14 & 60\end{bmatrix}\] = -5
\[\begin{bmatrix} 1 & 5\\ 2 & 6\end{bmatrix}\] = -4
\[\begin{bmatrix} 1 & 0\\ 2 & 1\end{bmatrix}\] = 1
Now, the new matrix is
\[\begin{bmatrix} -24 & -18 & 5 \\ -20 & -15 & 4\\ -5 & 14 & 1\end{bmatrix}\]
Creating the Matrix of Cofactors
To formulate the adjoint or the adjugate matrix, reverse the sign of the alternating terms as shown below:
We get the new matrix as
A = \[\begin{bmatrix} -24 & -18 & 5 \\ -20 & -15 & 4\\ -5 & 14 & 1\end{bmatrix}\]
Adj (A) = the new matrix is
A = \[\begin{bmatrix} -24 & -18 & 5 \\ -20 & -15 & 4\\ -5 & 14 & 1\end{bmatrix}\] X \[\begin{bmatrix} + & - & + \\ - & + & +\\ + & - & +\end{bmatrix}\]
Adj(A) = \[\begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4\\ -5 & 4 & 1\end{bmatrix}\]
Finding the inverse of 3 x 3 matrix
Now, if we will substitute the value of det (A) and the adj (A) in the formula:
\[A^{-1}\] = (1/det (A)) Adj (A)
\[A^{-1}\] = (1/1) = \[\begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4\\ -5 & 4 & 1\end{bmatrix}\]
Hence, the inverse of the given matrix is
\[A^{-1}\] = (1/1) = \[\begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4\\ -5 & 4 & 1\end{bmatrix}\]
Solved Examples
1. Find the Inverse of the Following Matrix
1 \[\begin{bmatrix} 2 & 1 & 1 \\ 1 & 1 & 1\\ 1 & -1 & 2\end{bmatrix}\]
Solution:
\[ -2\begin{bmatrix} 1 & 1\\ -1 & 2\end{bmatrix} -1\begin{bmatrix} 1 & 1\\ 1 & 2\end{bmatrix} +1\begin{bmatrix} 1 & 1\\ 1 & -1\end{bmatrix} \]
|A | = 2 (2-(-1)) -1 (2-1) +1(-1-1)
= 2 (2+1) -1 (1) +1 (-2)
= 2 (3) -1-2
= 6 - 3
= 3
|A | = 3
As A is a non-singular matrix. \[A^{-1}\] exists
Minor and cofactors of row 1
Minor of 2
\[\begin{bmatrix} 1 & 1\\ -1 & 2\end{bmatrix}\]
= (2-(-1))
= (2 + 1)
= (3)
= 3
Cofactor of 2 = + (3)
= 3
Minor of 1
\[\begin{bmatrix} 1 & 1\\ 1 & 2\end{bmatrix}\]
= (2-1)
= (2)
= 2
Cofactor of 1 = (-1)
= -1
Minor of 1
\[\begin{bmatrix} 1 & 1\\ 1 & -1\end{bmatrix}\]
=(-1-1)
= (-2)
= -2
Cofactor of 1 = + (-2)
= -2
Minor and cofactors of row 2
Minor of 1
\[\begin{bmatrix} 1 & 1\\ 1 & -2\end{bmatrix}\]
= (-2-1)
= -3
Cofactor of 1 = - (-3)
= -3
Minor of 1
\[\begin{bmatrix} 2 & 1\\ 1 & 2\end{bmatrix}\]
= (4 -1)
= (3)
= 3
Cofactor of 1 = - (-3)
= -3
Minor of 1
\[\begin{bmatrix} 2 & 1\\ 1 & -1\end{bmatrix}\]
(-2-1)
= (-3)
= -3
Cofactor of 1 = - (-3)
= 3
Minor and Cofactors of Row 3
Minor of 1
\[\begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatrix}\]
= (1-1)
= 0
Cofactor of 1 = + (0)
= 0
Minor of -1
\[\begin{bmatrix} 2 & 1\\ 1 & 1\end{bmatrix}\]
= (2-1)
= 1
Cofactor of -1 = - (1)
= -1
Minor of 2
\[\begin{bmatrix} 2 & 1\\ 1 & 1\end{bmatrix}\]
= (2-1)
= 1
Cofactor of 2 = + (1)
= 1
Cofactor Matrix
\[\begin{bmatrix} 3 & -1 & -2 \\ -3 & 3 & 3\\ 0 & -1 & 1\end{bmatrix}\]
Adjoint Matrix
\[\begin{bmatrix} 3 & -3 & 0 \\ -1 & 3 & -1\\ -2 & 3 & 1\end{bmatrix}\]
Hence, the inverse of the given matrix is
\[A^{-1}\] = \[\frac{1}{3}\] \[\begin{bmatrix} -3 & 3 & 0 \\ 1 & -3 & 1\\ 2 & -3 & -1\end{bmatrix}\]
1. Find the Inverse of the Following Matrix
\[\begin{bmatrix} 6 & 2 & 3 \\ 3 & 1 & 1\\ 10 & 3 & 4\end{bmatrix}\]
Solution:
\[2\begin{bmatrix} 1& 1\\ -1& 2 \\\end{bmatrix}\] \[-1\begin{bmatrix} 1 & 1\\ 1 & 2\end{bmatrix} + 1 \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\]
|A | = 2 (2-(-1)) -1 (2-1) +1(-1-1)
= 2 (2+1) -1 (1) +1 (-2)
= 2 (3) -1-2
= 6 - 3
= 3
|A | = 3
As, A is a non-singular matrix. \[A^{-1}\] exists
|A| = 6 (4-3) - 2 (12-10) + 3 (9-10)
= 6 (1) - 2 (2) + 3 (-1)
= 6 - 4 - 3
= 6 - 7
= -1
|A| = -1 ≠ 0
Minor and Cofactors of Row 1
Minor of 6
\[\begin{bmatrix} 1 & 1\\ 3 & 4\end{bmatrix}\]
= (4-3)
= (1)
= 1
Cofactor of 6 = + (1)
= 1
Minor of 2
\[\begin{bmatrix} 3 & 1\\ 10 & 4\end{bmatrix}\]
= (12-10)
= (2)
= 2
Cofactor of 2 = (-2)
= -2
Minor of 3
\[\begin{bmatrix} 3 & 1\\ 10 & 3\end{bmatrix}\]
(9-10)
= (-1)
= -1
Cofactor of 3 = + (-1)
= -1
Minor and Cofactors of Row 2
Minor of 3
\[\begin{bmatrix} 2 & 3\\ 3 & 4\end{bmatrix}\]
= (8-9)
= (-1)
= -1
Cofactor of 3 = - (-1)
= 1
Minor of 1
\[\begin{bmatrix} 6 & 3\\ 10 & 4\end{bmatrix}\]
= (24-30)
= (-6)
= -6
Cofactor of 1 = + (-6)
= -6
Minor of 1
\[\begin{bmatrix} 6 & 2\\ 10 & 3\end{bmatrix}\]
(18-20)
= (-2)
-2
Cofactor of 1 = - (-2)
= 2
Minor and Cofactors of Row 3
Minor of 10
\[\begin{bmatrix} 2 & 3\\ 1 & 1\end{bmatrix}\]
= (2-3)
= -1
Cofactor of 1 = + (-1)
= (-1)
= -1
Minor of 3
\[\begin{bmatrix} 6 & 3\\ 3 & 1\end{bmatrix}\]
= (6-9)
= (-3)
= -3
Cofactor of 3 = - (-3)
= 3
Minor of 4
\[\begin{bmatrix} 6 & 2\\ 3 & 1\end{bmatrix}\]
= (6-6)
= (0)
= 0
Cofactor of 4 = + (0)
= 0
Cofactor Matrix
\[\begin{bmatrix} 1 & -2 & -1 \\ -1 & -6 & 2\\ -1 & 3 & 0\end{bmatrix}\]
Adjoint of Matrix
= \[\begin{bmatrix} 1 & 1 & -1 \\ -2 & -6 & 3\\ -1 & 2 & 0\end{bmatrix}\]
\[A^{-1}\] = 1/1 = \[\begin{bmatrix} -1 & -1 & 1 \\ 2 & 6 & -3\\ 1 & -2 & 0\end{bmatrix}\]
FAQs on Inverse of 3 by 3 Matrix
1. What is the purpose of the Inverse of 3 by 3 Matrix in Algebra?
The inverse of the 3 by 3 Matrix is highly beneficial in Algebraic problems that are highly complicated and needs several steps to be solved. For example, if there is a problem that needs you to divide by a fraction then you can more easily multiply by its reciprocal instead of using the fraction to divide the number. This is called the inverse operation. Similarly, since there is no presence of division that needs to be done in a matrix all you need to do is use the inverse operations and multiply with the help of it. Calculating the Inverse of 3 by 3 Matrix is a very hard task but you can understand and practice them for a better understanding.
2. How to understand the Inverse of 3 by 3 Matrix well with just solved examples?
You don't need to understand the concept involved in the Inverse of 3 by 3 Matrix by just solving the solved examples. Being a student you might have access to the textbook which consists of problems and their related theory. First, make sure you refer to the textbook and check what is provided in it to understand the basic concepts and what the numbers and brackets involved in the Inverse of 3 by 3 Matrix indicate. If you do not have access to the textbook then you can refer to the Vedantu NCERT textbooks provided. Once you understand the concept, then move on to the solved examples that are present in the textbook.
3. Does Vedantu only provide Inverse of 3 by 3 Matrix solved examples or does it also provide other important study materials?
Yes, Vedantu does provide a lot of other study materials other than Inverse of 3 by 3 Matrix solved examples and hence students who do not have access to other extra study materials do not need to worry as they can access all the facilities for free of cost on Vedantu. Students can also get access to the Vedantu sample papers that may contain examples that are similar to Inverse of 3 by 3 Matrix and are deemed to be important for exams.
4. Is it possible to use a calculator to solve questions that are related to the Inverse of 3 by 3 Matrix?
Yes it is possible to use a calculator to solve the problems related to Inverse of 3 by 3 Matrix and the steps can be given as follows
First, select a calculator that consists of matrix capabilities
enter the matrix onto your calculator
Now select the edit submenu given
Select a name for the matrix you have entered
enter the dimension of the matrix entered
enter each of the elements of the matrix
quit the matrix function and use the inverse key to find the inverse matrix from the original matrix
convert the inverse matrix to get the answer to the question.
While it is easy to find the answers using a calculator, not using the calculator is highly recommended so that you understand the concept well.
5. How to get easy shortcuts to solve the Inverse of 3 by 3 Matrix sums that are provided in the Algebra textbook?
Problems based on the Inverse of 3 by 3 Matrix might seem like they are too hard to understand and you need a shortcut so that you can solve them in exams. However, it is seen that there are no shortcuts when it comes to solving Math problems. You not only need to understand the concept but you will also need to understand the steps that are involved while solving the question. Even though it may take some time and practice it is definitely worth all the marks that you score once you solve them in the exam.