Intuitive and geometric meaning of Matrix Inverse or inverse of Matrix

For defining Inverse of Matrix Mathematically , consider A as any square matrix then  A^{-1} which is called inverse of Matrix A , is square matrix which satisfies the following equation.

 A*A^{-1}=A^{-1}*A=I

where I is identity matrix.

This definition was would satisfy an average student. However for above average student who really want to have a feel of it. There are may questions like. What is geometrical meaning or intuitive meaning of Inverse of matrix.

If you are more curious about understanding and feeling inverse of matrix. Then read the following content or watch this video which gives very nice interpretation, geometrical meaning and intuitive feel of Inverse of Matrix.

After lot of googling I found this video for which explains it in very nice way. I can give you summary of what is there in video however reproducing it in text would make an answer length. If you want to know it anyways you can always watch that video. So lets start with some simple number.

For a number x,  x^{-1} is defined as  \frac{1}{x}

However this theory falls flat when it comes to matrix

Let us say that Matrix A is defined as

 A= [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]

then why

 A^{-1} \neq [\begin{matrix} \frac{1}{1} & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{4} \end{matrix}]

This question used to boggle my me every time I come across Inverse of Matrix. I asked this question to many of my friends but no one was able to satisfy my doubt. Even Math’s Professor was also not able to give answer to my doubt. My B-Tech college was not that good, so I didn’t expect my Math’s Prof to give answer to it.

Then finally all my doubts was clear when I came across this video on Internet. To explain concept of inverse of matrix, he started right from concept of inverse of number. Then he explained why we define  x^{-1} = \frac{1}{x}

I was knowing this from my school days but his interpretation of inverse was quite awesome. I would like you to see that video if you have time. Actually it is a bit lengthy but explains inverse of matrix in very nice way.

If you don’t have time to watch video then I will put forward meaning of inverse in simple words. Although i am not good at explaining this I will try my best.

Any matrix can be assumed as a means to transform vector. Suppose we define matrix A. Then if it operates on any vector X like A.X = B, then it transforms vector X into new vector B. Now inverse of matrix is another matrix which transform Vector B back to Vector X. That matrix which transforms B back to X is called inverse of matrix A

Mathematically  A*A^{-1}=I

6 thoughts on “Intuitive and geometric meaning of Matrix Inverse or inverse of Matrix”

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