If a b c ≠0, find the inverse of [ a 0 0 0 b 0 0 0 c

If a b c ≠0, find the inverse of [ a 0 0 ...

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, results in the identity matrix. This new matrix is called the inverse. It is an essential concept in linear algebra, particularly in solving systems of linear equations and understanding the properties of linear transformations.

Diagonal Matrix

A diagonal matrix is a type of matrix where all the entries outside the main diagonal are zero. This structure simplifies many operations, such as matrix multiplication and inversion, as the behavior is primarily governed by the individual diagonal entries.

Invertibility Condition

For a matrix to be invertible, it must be non-singular, meaning its determinant is nonzero. In the case of a diagonal matrix, this condition is satisfied if and only if all the entries on the main diagonal are nonzero, allowing each to be individually inverted.

Transcript

00:01 In the question given matrix is that is elements are in the first row, first is that is a00 for first row and second row elements are that is 0 b 0 and for third row elements are 0 0 c so this matrix is given in the cushion now i will representing it by a.

00:30 Now in the question it is saying that abc is not equal to 0 and we have to find inverse of given matrix means i have to find a inverse.

00:46 So this data is given in the question.

00:49 So for finding a inverse what i do is first i will write the given matrix in augmented form.

00:56 So it will be written as on the left side means this side the we have to write the given matrix and and on the right side we have to write the identity matrix with same order so so it so it means it will be written as that on the left side that will be a zero zero zero and zero c so these are elements written in the row respectively and on the right sides the elements are in the first row will be that is one zero zero and in the second row elements are 010 and in the third row the elements are 001 now next step is we we will do some operations to make the left side similar to right side means making this left side as a identity matrix so doing this the right side also changed by when i am when we take operations so it means doing operations on both sides the both side elements will change so in the result on the left side it will become identity matrix and on the right side it will become inverse matrix so that's a start by doing first operation that is first operation will be applied for first row that is r1 divided by a and second operation that is applying for second row that is r2 divided by b and and third operation, row operation that is applying for third row that is r2 divided by r3 divided by c...

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