2 1 1 If A= 1 2 1 1 1 2

Name: Problem 16, Chapter 4: NCERT Class 12 Part 1 - Math
Uploaded: 2021-06-18T09:27:10-08:00
Duration: 13 min 44 s
Channel: Suman Saurav Thakur
Description: 2 1 1 If A= 1 2 1 1 1 2

2 1 1 If A= 1 2 1 1...

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are scalars and corresponding nonzero vectors that satisfy the equation A*v = ?*v for a matrix A. They capture many intrinsic properties of the matrix, such as its scaling behavior along specific directions. For symmetric matrices, eigenvalues are guaranteed to be real and eigenvectors can be chosen to form an orthonormal basis, which is crucial in many applications like principal component analysis and solving differential equations.

Determinant and Characteristic Polynomial

The determinant of a matrix is a scalar value that summarizes certain properties of the matrix, including whether the matrix is invertible (a nonzero determinant indicates invertibility). In the context of eigenvalue problems, the determinant is used to form the characteristic polynomial by setting det(A - ?I) equal to zero, allowing for the computation of eigenvalues, which are fundamental in understanding the behavior of linear transformations.

Symmetric Matrices

A symmetric matrix is one that is equal to its transpose, meaning that the element in the i-th row and j-th column is the same as the element in the j-th row and i-th column. This property often simplifies many computations and leads to important theoretical results, such as real eigenvalues and orthogonal eigenvectors, which are particularly useful in optimization and quadratic forms.

Matrix Representation

A matrix is a rectangular array of numbers arranged in rows and columns that can represent linear transformations, systems of equations, or data. In the context of linear algebra, matrices provide a systematic way to perform operations like addition, multiplication, and inversion, and serve as a compact representation of complex mathematical ideas.

Transcript

00:01 We have log number 16 and with a matrix being given a equal to 2 minus 1 1 minus 1 2 and minus 1 and 2 and 1 minus 1 and 2 and we need to verify that a q minus 6a square plus 9a minus 4 i 0 and then find a inverse okay for that we have to find 1 a square then a cube a square is a into a so let us write the matrices 2 minus 1 1 minus 1 2 minus 1 2 in 2 same thing 2 minus 1 1 minus 1 1 minus 1 and 2 so it will be 2 into 2 into 2 into 2 2 into 2 4 plus 1 because we are multiplying like this 2 plus 1 plus 1 plus 1 plus 1 4 plus 1 plus 1 so and a minus 2 minus 2 minus 1 then 2 plus 1 plus 2 and then minus 2 minus 2 minus 1 plus 1 minus 2 and minus 2 and minus 2 1 plus 1 plus 1 plus 4 so a squared becomes equal to 2 2 2 plus 1 plus 2 and minus 2 and minus 2 1 plus 1 plus 4 so a squared becomes equal to six minus five five minus five and six and minus five this is five minus five and six this is five minus five and six okay so a cube is a square into a a square into a so a square is 6 minus 5 5 and then minus 5 6 minus 5 5 5 5 minus 5 6 in 2 a is 2 minus 1 1 minus 1 2 this is 6 into 12 plus 5 plus 5 minus 6 plus 10 plus 10 we are multiplying like this minus six plus 10 sorry minus 10 minus 10 and then minus 5 6 plus 5 plus 10 minus 10 minus 5 then 5 plus 6 minus 5 then minus 5 minus 10 minus 6 5 plus 5 plus 12 okay so a q becomes equal to 22 minus 21 21 minus 21 22 and minus 21 okay 21 21 21 minus 21 and 22 this is a cute now we will be plugging in the equation equation given is a cube minus 6 a square 6a square plus 9a minus 4 i so let's us of the equation is this let us start talking the values 22 minus 21 21 minus 21 22 minus 21 22 minus 21 21 minus 21 22 minus 6 a square a square has the value 6 minus 5 and 5 minus 5 6 minus 5 5 and 6 plus 9 a has the value given value 2 minus 1 1 minus 1 and 2 minus 1 and 2 and then minus 4 a so minus 4 one zero zero one zero one zero zero and one okay so this will be equal to 22 plus 36 plus 18 because this is 22 plus 6 we multiplied with each of the member so 36 plus 18 plus 4 okay so this comes out to be 22 plus 36 plus 18 minus equation was minus 6a square sorry minus 6a square was the equation so it should be minus and then minus 4a so it should be minus 2 okay so 22 minus 36 minus 36 plus 18 minus 4 this is minus 21 plus 30 minus 9 it will be 0 third is 21 minus 30 21 minus 30 plus…

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