The Inverse of a Square Matrix

<aside> 💡 Let $A$ be a $n\times n$ matrix. The inverse of $A$ is another $n\times n$ matrix $B$ such that $AB=BA=I_{n\times n}$ and is denoted by $A^{-1}$.

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Uniqueness of Inverse

• Let us assume that the inverse of a matrix is not unique.
• i.e., $AB=BA=I$ , also $AC=CA=I$, and $B\neq C$.
• Now , Let us take $CAB.$
• By the property of associativity, $C(AB)=(CA)B$→(1)
• Now, $AB = I ,$ so (1) becomes $CI=(CA)B \implies C=(CA)B$→(2)
• Also, $CA = I,$ so (2) becomes $C=IB \implies$ $C=B$
• Hence Inverse of a matrix is unique.

Inverse of $A$ exists $\implies$$det(A)$ has to be non-zero. (since $det(A^{-1})=\frac 1 {det(A)}$) The converse is also true.

The Adjugate of a Square Matrix

<aside> 💡 The adjugate matrix of $A$ is defined as: $adj(A)=C^T$, where $C$ is the cofactor matrix whose $(i,j)^{th}$ entry is $C_{ij}$.

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Adjugate and Inverse

If $A$ is an $n\times n$ matrix and $det(A)\neq0$, then $A^{-1}$ exists and:

<aside> 💡 $A^{-1} = \frac 1 {det(A)} ~adj(A)$

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