Question

find the inverse of matrix |123;245;356| by elementary row operation​

Answer 1

answer is in the images with reason and calculation

Answer 2

Answer:

The answer to this question is.     $\left[\begin{array}{ccc}1&2&3\\2&4&5\\3&5&6\end{array}\right]$

Step-by-step explanation:

Matrix Inverse

If A is a non-singular square matrix, there is an existence of n x n matrix A-1, which is called the inverse matrix of A such that it satisfies the property:

AA-1 = A-1A = I, where I am the Identity matrix

The identity matrix for the 2 x 2 matrix is given by

Where a, b, c, and d represent the number.

The determinant of matrix A is written as ad-bc, where the value of the determinant should not equal zero for the existence of an inverse. The inverse matrix can be found for 2× 2, 3× 3, …n × n matrices. Finding the inverse of a 3×3 matrix is a bit more difficult than finding the inverses of a 2 ×2 matrix.

Inverse Matrix Method

The inverse of a matrix can be found using the three different methods. However, any of these three methods will produce the same result.

Method 1:

inverse-matrix-method

Similarly, we can find the inverse of a 3×3 matrix by finding the determinant value of the given matrix.

Check out: Inverse matrix calculator.

Method 2:

One of the most important methods of finding the matrix inverse involves finding the minors and cofactors of elements of the given matrix. Observe the below steps to understand this method clearly.

The inverse matrix is also found using the following equation:

A-1= adj(A)/det(A),

         where adj(A) refers to the adjoint of a matrix  ,and  det(A) refers to the determinant of a matrix A.

The adjoint of a matrix A or adj(A) can be found using the following method.

         Too find the adjoint of a matrix A first, find the cofactor matrix of a given matrix and then  

         take the transpose of a cofactor matrix.

The cofactor of a matrix can be obtained as

Cij = (-1)i+j det (Mij)

Here, Mij refers to the (i,j)th minor matrix after removing the ith row and the jth column. You can also say that the transpose of a cofactor matrix is also called the adjoint of a matrix A.

Learn how to find the adjoint of a matrix here.

Similarly, we can also find the inverse of a 3 x 3 matrix. Here also the first step would be to find the determinant, followed by the next step – Transpose.

Now from the above method we calculate transpose of the given matrix which would be.

   $\left[\begin{array}{ccc}1&2&3\\2&4&5\\3&5&6\end{array}\right]$

Now dividing it with its determinant we got the same matrix as its determinant is 1.

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