Matrix Inversion

Inverse definition

An (𝑛×𝑛)(𝑛 \times 𝑛) square matrix 𝐴𝐴 is called invertible (also nonsingular or nondegenerate) if there exists an (𝑛×𝑛)(𝑛 \times 𝑛) square matrix 𝐡𝐡 such that:

𝐴𝐡=𝐡𝐴=𝐼𝑛𝐴𝐡 = 𝐡𝐴 = 𝐼_𝑛

  • Here, 𝐼𝑛𝐼_𝑛 denotes the (𝑛×𝑛)(𝑛 \times 𝑛) identity matrixa diagonal matrix where each element in the diagonal is equal to one. where we use ordinary matrix multiplication.

  • If the matrix is invertible, then the matrix 𝐡𝐡 is uniquely determined by 𝐴𝐴 and is called the inverse of 𝐴𝐴, denoted by π΄βˆ’1𝐴^{βˆ’1}.

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