how to find inverse of matrix

If you have a matrix equation to solve, you should use the simple technique to take the inverse of the given matrix. There are a few ways to find the inverse of the matrix; it’s a little tricky because some matrices do not have an inverse. Before learning how to take the inverse of a 3×3 matrix, you have to take determinants first, what is the transpose of a matrix and the inverse formula, and if a matrix has an inverse or not. There are no short or quick methods to find an inverse of a matrix, and you have to follow all the rules given below to find an inverse of a 3×3 matrix.

Find the inverse of the following 3×3 matrix.


Name it as A, and you have to find A–1 of it. And the formula of an inverse is:


First of all, find the determinant of the matrix, and it is symbolically represented as det(A). If the determinant’s answer is zero, this equation has no inverse, and your question is finished.


To get the cofactors of matrix A, calculate the 2×2 determinant as shown in the picture below. Calculate the minor matrix and cofactors of each Row quite carefully.


Now, create the matrix of cofactors and assign signs to all the elements of the matrix. Follow the addition sign rule, such as;

  • +* + = + and – * – = –
  • +* – = + and – * + = – (in these two cases, always keep the sign of the larger value)
adjoint 1, 2

Now transpose the matrix you have created so far. Transpose of a matrix is a flipped version of your original matrix, where you swap the rows with columns.
Like: Adj (A) = [ cofactor (A)]T

See the attached image quite carefully, and then you will understand the changed position of rows and columns.


The matrix formed after changing signs is called the Adjugate matrix, and symbolically it is represented as Adj (A)

Now apply the formula and perform the further steps needed to simplify the equation.

apply-formula 1, 2

And here is your inverse of the 3×3 matrix.

answer a inverse

Other ways to find the inverse of a matrix

The best thing about practising Math is that there are always more than one ways to solve a problem correctly. Moreover, to find the 3×3 inverse of a matrix, use the linear row reduction or elementary row operation method and a scientific calculator. Elementary row operation is a bit complicated method and is used relatively less by the students. On the other hand, use a calculator with matrix capabilities to find an inverse or perform other matrix equations. Also, the calculator will not give you the exact answer; it is a machine, after all. You have to convert the decimal answer of the calculator into a fractional form on your own. However, some calculators have the option of converting decimal into fraction form.

Math is the fundamental subject that is mainly ranked as a challenging course among students. However, the fact is, this subject comes with so many methods and techniques, and it depends on the Maths tutors or teachers how they transfer the concepts to its students and help them improve in Maths. Just like that, each teacher has their way of calculating the inverse of the 3×3 matrix.

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