Find the inverse of the matrix (if it exists). [ 2 0 0 0 3 0 0

Find the inverse of the matrix (if it exists). [ 2 0...

Key Concepts

Diagonal Matrix

A diagonal matrix is a type of matrix in which all the non-diagonal entries are zero. Diagonal matrices are particularly simple in terms of their algebraic properties, as operations such as finding the inverse are straightforward—each nonzero diagonal element can be replaced by its reciprocal. This simplicity underlies many techniques used to simplify and analyze linear transformations.

Invertibility Condition

A square matrix is invertible if and only if its determinant is nonzero. This condition guarantees that the matrix does not have any row or column dependencies and that a unique solution exists for linear systems involving the matrix. Checking the determinant is a common method to determine if an inverse exists.

Matrix Inverse

The matrix inverse of a square matrix A is another matrix, denoted A?¹, such that when it is multiplied with A, the result is the identity matrix. Not every matrix has an inverse; the existence of an inverse is crucial in solving systems of linear equations, among other applications. The inverse is essential in various areas of linear algebra and applied mathematics.

Transcript

00:01 Hello students we are given a matrix we just need to find the value of inverse if it exists so how we will find first of all we are supposed to know that we can consider it as matrix a so matrix a is determinant is nothing but two times of okay three five j of 15 minus zero after the minus zero plus zero so basically 15 two is 30 it's all about the determinant of a now we are supposed to know that the formula of a inverse can be written as adjoint of a divided by determinant of a and adjoint of a it is nothing but the transpose of the matrix of the co -factor so basically we can write it as first column can be written as c1 c1 c1 2 2 c2 c2 3 3 3 3 okay it's all about our adjoint of a now we are supposed to know that what is c ij is nothing but minus 1 to the power i plus j times of m ij okay so basically what we can do here we are supposed to know that we just need to write matrix a again so it is nothing but two zero zero here zero three zero and here zero zero five okay it's all about our matrix a we can say again you can just write two zero zero zero three okay zero five it's all about our matrix a now we are supposed to know that we just need to calculate c one one c factors okay who factors c one one two c one three how we will find see we can say that it will be positive of m one one okay and it is minus one to three the power 1 plus 2 comes out as minus 1 to the power 3 from this formula so basically minus it will be negative of m12 okay and it is it can be written as directly positive of m13 now what we can do here m1 how we will find 3 5 jar 15 minus 0 comes out as 15 here minus of m12 we just need to skip this this row in this column what we'll get 0 minus 0 comes out as again 0 here m13 0 minus 0 okay so again now c21 c22 and c23 how we will find those values see we can say that we just need to skip this column and this row what we'll get zero minus zero comes out as again zero and sorry it…

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