Find the adjoint of the matrix A . Then use the adjoint to find the inverse of A (if possible).

Find the adjoint of the matrix A . Then use the adjoint to...

Key Concepts

Adjugate Matrix

The adjugate (or classical adjoint) of a square matrix is the transpose of the cofactor matrix. It plays a crucial role in various matrix computations, especially in finding the inverse of a matrix. The process involves calculating the cofactor for each element of the matrix, arranging these cofactors into a new matrix, and then taking the transpose of that matrix.

Cofactor

A cofactor is obtained by multiplying the minor of an element by a sign factor that alternates according to the position of the element (determined by (-1)^(i+j)). The minor is the determinant of the submatrix formed when the row and column corresponding to the element are removed. Cofactors are used to form the cofactor matrix, which is central to computing the adjugate matrix.

Determinant

The determinant is a scalar value that summarizes certain properties of a square matrix, including its invertibility. It is calculated through recursive expansion, usually by minors and cofactors, and its nonzero value is a necessary and sufficient condition for the existence of an inverse. The determinant also plays a key role in the formula for finding the inverse of a matrix using the adjugate.

Matrix Inversion Using Adjoint

Matrix inversion using the adjoint involves computing the adjugate of the matrix and dividing it by the determinant of the matrix. This method highlights the relationship between the matrix, its cofactor matrix, and the determinant. It is applicable to any invertible square matrix, and the formula is given by A^(-1) = (1/det(A)) * adj(A), provided that the determinant is not zero.

Transcript

00:01 Let's friends we are given a matrix and we need to find first of all the adjoint of a and if the matrix is a determinate we need to find the a inverse as well as if the a inverse exists okay so first of all we are supposed to know that the what is the formula of adjoint of a adjoint of a adjoint of a can be written as in the pattern of cofactors just like that see c1 c1 c1 okay and the another column will be written as c21 c2 and last will be c23 last column can be written as c31 c32 and c33 okay it's all about the adjoint of a so first of all we are supposed to know that c i j it is co -factor formula okay cij is nothing but minus 1 to the power i plus j times m ij is nothing but the minor of every each and every element of the matrix okay so first of all we will go with the c11 c1 can be written as minus 1 to the power 1 plus 1 which means minus 1 to the power 2 which comes out as positive so positive 1 times of m11 can be written as m11 1 1.

01:04 So basically we will find m1 1 how we will find 1 times of 2 is nothing but 2 minus 2 times of minus 1 is nothing but minus 2 so minus minus plus and 2.

01:14 So 2 plus 2 is nothing but 4.

01:15 Now c12.

01:17 Okay students c 1 2 can be calculated as minus of m 1 2.

01:21 Okay so basically minus of m 1 2 can be written as 0 times or 2 is nothing but 0 minus 2 times of minus 1 is nothing but minus 2.

01:30 So basically minus minus plus and it will be two okay sorry first of all we will solve basic term zero minus of okay two times of minus one is nothing but minus two so basically minus two so minus minus plus and it will be two so minus of two is nothing but minus two now c -13 how we will calculate it can be written as m13 okay so m13 can be solved as zero times of two is nothing but zero minus two times of one is nothing but two so basically 0 minus 2 can be written as minus 2 okay now we need to find the c21 c22 and c 2 3 okay so how we will find c 2 plus 1 is nothing but 3 so minus it can be written as minus of m 2 1 it can be written as m 2 2 2 it can be written as minus of m 2 3 okay now what we need to do we just need to calculate minus of m 2 1 m 2 1 can be calculated as we just need to skip that column and that row so we can write it as two times of two is nothing but four okay minus two times of three is nothing but six okay so basically it will be two now we can calculate the c22 sorry m2222 can be calculated by skipping that column and that row we will get 1 times of 2 is nothing but 2 okay minus 1 times of 2 is nothing but 2 minus 2 times of 3 is nothing but 6 so basically 2 minus 6 we can write it as 2 minus 6 which comes out as minus 4 now minus of m2 3 m2 3 can be calculated by skipping this column in this row we will get 1 times of 2 is nothing but 2 okay minus 2 2 times of 2 is nothing but 4 so basically 2 minus 4 it is minus of minus 2 which comes out as plus 2 now we need to find those last column terms will be can be written as c3 1 c3 2 and last will be c 3 3 3 3 3 plus 1 is nothing but 4 which is e 1 so we can write it directly as m 3 1 it is 3 plus 2 5 so odd number, so basically minus of m3 2 and it is it will also positive say m 3 3 okay now how we will calculate see m 3 1 can we calculate by skipping this column and this row? what we will calculate? what we will calculate? we just can see that two times of minus 1 is nothing 1 minus 2 minus 1.

03:49 Minus 1 times of 3 is nothing 1 3 sorry minus 3 so basically minus 2 minus 3 okay see minus 2 and minus 3 3.

03:56 So basically minus 2 and minus 3 it is minus 5 so basically minus 2 minus 3 can be written as minus 5 now minus of m 3 2 m 3 how we will calculate m 3 2 we just need to skip that column and that row you will get 1 times of minus 1 is nothing but minus 0 so basically minus 1 minus 0 so basically minus 1 minus 0 okay minus 1 minus 0 it is minus minus plus so basically it will be 1 now m 3 how we will calculate we just need to know that we need to skip that particular row, sorry, column and that row.

04:28 What will get 1 times of 1 is nothing but 1 minus 0 times of 2 is nothing but 0.

04:32 So basically 1 minus 0.

04:34 We can write 1 minus 0 can be written as 0.

04:37 Sorry, can be written as 1.

04:39 Okay...

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