Matrix and inverse matrix question?

In summary, the conversation is about finding the inverse of a matrix A using the equation A^2 = λA - 2I, without using the adjoint method. The solution involves algebraically manipulating the equation and factoring out an A to find the multiplicative inverse of A.
  • #1
nishantve1
76
1

Homework Statement


how do I solve this question
https://www.dropbox.com/s/bn958xnm5483s6u/photo%20%281%29.JPG
(This link if the image is not visible : https://www.dropbox.com/s/bn958xnm5483s6u/photo (1).JPG)
just so the equation is not clear it says A2 = λA - 2I

The inverse should be found through the equation in the question and not through the adjoint method

Homework Equations


The Attempt at a Solution



The equation says
A2 = λA - 2I
So, I mulitplied by A-1
This gives
A2 A-1 = λA A-1 - 2IA-1
{A-1 . A = I}

now the equation becomes
A = λI - 2A-1

I am stuck here since I cannot simply find A-1 by the adjoint method this equation has two unknowns A-1 and λ . Or maybe I am misinterpreting the question IDK
 
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  • #2
But [itex]\lambda[/itex] is not an unkown.

You are given a very specific matrix A. From your specific matrix A, and from your equation [itex]A^2= \lambda A- 2I[/itex], you can deduce what [itex]\lambda[/itex] is.
 
  • #3
How did I miss that . Thanks
 
  • #4
I see no point in looking for specific values of [itex]\lambda[/itex]. To find the multiplicative inverse of A:
1) algebraically manipulate the equation, [itex]A^2= \lambda A- 2I[/itex], to get "I" alone on the right.
2) factor out an "A".
 

FAQ: Matrix and inverse matrix question?

What is a matrix?

A matrix is a rectangular array of numbers, arranged in rows and columns. It is used to represent a set of equations or to perform operations such as addition, subtraction, and multiplication.

What is an inverse matrix?

An inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is denoted by A^-1 and can be used to solve equations involving matrices.

How do you find the inverse matrix?

The inverse of a matrix can be found by using the Gauss-Jordan elimination method or by using the adjugate matrix formula. It is important to note that not all matrices have an inverse.

What is the significance of an inverse matrix?

An inverse matrix is significant because it allows us to solve equations involving matrices, which are often used in fields such as engineering, physics, and economics. It also has applications in computer graphics and cryptography.

Can any matrix have an inverse?

No, not all matrices have an inverse. For a matrix to have an inverse, it must be a square matrix (number of rows = number of columns) and its determinant must be non-zero. If the determinant is zero, the inverse does not exist.

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