Inverse Matrix

from University of Tokyo 1S1 Mathematical Sciences Foundation: Linear Algebra Inverse Matrix

• In the world of numbers, the inverse of a number refers to a reciprocal.
  • Reciprocal: When $ax=1$, $a$ is the inverse matrix of $x$.
  • Inverse matrix: When $AX=XA=E$ (Identity Matrix), A is the inverse matrix of X.
    • It is actually quite nontrivial that AX=XA for an inverse matrix because it does not follow the Commutative Law.
  • The inverse matrix is denoted as $A=X^{-1}$.
    • It is not written as a fraction. Interesting, huh? (blu3mo)
  • Not all matrices have an inverse matrix.
    • Some of them are Invertible.
    • The determinant Matrix Determinant is used to determine this.
    • It is interesting to perform a Linear Transformation with a matrix that does not have an inverse matrix (takker).
      • To better understand this and other concepts related to matrices, it is highly recommended to study the book ”Matrix for 7-Year-Olds: Understanding through Visualization (Bluebacks)“.
      • Similarly, the book ”Math Girl’s Secret Notes: What Matrices Draw” provides a clear understanding of these concepts.
• Matrix Determinant
  • Contains various information.
• If detA ≠ 0, then an inverse matrix exists.
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