(ii) the matrix X satisfying A^-1 XA=B (iii) the matrix A using A^-1

Name: Problem 3, Chapter 2: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-13T15:08:54-08:00
Duration: 14 min 21 s
Channel: Anas Venkitta
Description: (ii) the matrix X satisfying A^-1 XA=B (iii) the matrix A using A^-1

(ii) the matrix X satisfying A^-1 XA=B (iii) the matrix A...

Key Concepts

Similarity Transformation

A similarity transformation involves transforming a matrix into another matrix that represents the same linear transformation but in a different basis. This is typically expressed in the form A?¹XA = B, meaning that X is similar to B. The process of applying the inverse and then the original matrix corresponds to a change of basis, which preserves many properties of the original matrix such as its eigenvalues.

Solving Matrix Equations

Solving matrix equations often involves operations such as multiplying both sides of the equation by the appropriate inverse matrix. In equations like A?¹XA = B, one strategically uses the inverse to isolate the unknown matrix. By multiplying by A on the appropriate side, the equation can be rearranged to form expressions like X = ABA?¹, demonstrating the interplay between inversion and matrix multiplication in solving for unknown matrices.

Matrix Inverse

The matrix inverse is a fundamental concept in linear algebra. It is defined for a square matrix A such that there exists another matrix, denoted A?¹, which when multiplied by A (from either side) results in the identity matrix. This property is essential when solving matrix equations because it enables the isolation of unknown matrices and provides a method to 'undo' the multiplication by A.

Transcript

00:01 In the question we are told that matrices a and b satisfy the equation, satisfy the equation given as ab is equal to the inverse of b.

00:25 So a times b is equal to the inverse of b is what is the equation that matrices a and b satisfies.

00:33 And we are told that b is equal to 1 minus 2 1 0 so this is b and now we are told to first we are told to find the value of k if k times i which is the identity matrix is equal to 4 b square minus 4 b so this is what we need to find first we have to find first we have to find the value of k.

01:07 So what we will do over here is we can write k i is then equal to 4 times b squared which is 1 minus 2 1 0 times 1 minus 2 1 0 minus 2 1 0 minus 4 times 1 minus 2 1 0 0.

01:36 So now once solving this equation what we will get as k i, k times i is minus 8 0 minus 8.

01:48 And from this we can say that since i is equal to i is 1 001 we can say that.

02:04 Minus 8, right? since k times i should give us the matrix minus 8 0 minus 8.

02:12 So the only value that would give that is minus 8.

02:16 So that is the first question.

02:20 Next up what we are asked is that the matrix x is satisfying.

02:27 There is a matrix x that satisfies that satisfies that satisfies the equation a inverse times x times a as equal to the matrix b.

02:48 So this is what we have to, what we are given and now we have to find the matrix x, find the matrix x.

03:01 So over here what we can do is let's take the equation a inverse x times a equal to b and let's multiply with a on both sides so a inverse times a is equal to 1 so we will have x a is equal to ab and now we can write let's multiply a inverse on both sides and what we will have as x times a a minus equal to a times a b times a minus a inverse not a minus it is a inverse so now since a times a inverse is equal to i which is the identity matrix we can write x is equal to a b a times b times mine a inverse so this is what we have and now what we can do is since a b we are given in the question that ab ab ab that ab is equal to b inverse right so over here we have ab times a inverse so we will have ab as ab is equal to b inverse means this would be equal to b inverse times a inverse right and we can write this as we can write this as as a b inverse so it's a of matrices that when we take the inverse of the matrix the product of the matrix a times b it would be equal to b inverse times a inverse so this is a property that we need to keep in mind so now that we have wrote a b inverse we can again use the same rule that as same condition that has been given in the question that a b is equal to b inverse so again have b inverse the inverse of b inverse right which is equal to b itself so the required matrix a is matrix x is then equal to b which is 1 minus 2 0 1 0 so this is the required matrix x so now we have found the matrix x so let's move on to the next part of the problem problem that is the part 3 and in this part what we are told is we are told that we should find the matrix a find a using the inverse of a that is a inverse.

06:10 So what we do over here is we can take the condition from statement b that is a b is equal to b inverse and from this let's multiply b inverse on both sides so we will have b times b inverse is equal to i so we will have a and over here we have b inverse times b inverse which is b inverse squared right and now let's take the square of b a square of b right right so, from this since a is equal to b inverse squared, from this we can write a inverse is equal to b squared...

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