Invertible Matrices

• If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A-1 and A is the inverse of B
• Inverse of a matrix if it exists is unique.
• AA−1 = A−1A = In
• (A−1)−1 = A
• (kA)−1 = k−1A−1 if k ≠ 0
• (AB)−1 = B−1 A−1

View the Topic in this video From 24:58 To 48:27

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•  For a matrix A = \tt \begin{bmatrix}a & b \\c & d \end{bmatrix},\ adj\ A=\begin{bmatrix}d & -b \\-c & a \end{bmatrix}\ and\ A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d & -b \\-c & a \end{bmatrix} if ad − bc ≠ 0
• If A is a triangular matrix, then A−1, if it exists is a triangular matrix of the same kind.
In fact if \tt A=\begin{bmatrix}a_{11} & 0 & 0 \\a_{21} & a_{22} & 0 \\a_{31} & a_{32} & a_{33} \end{bmatrix} and a11 a22 a33 ≠ 0, then
\tt A^{-1}=\frac{1}{a_{11}a_{22}a_{33}}\begin{bmatrix}a_{22}a_{33} & 0 & 0 \\-a_{21}a_{33} & a_{33}a_{11} & 0 \\A_{13} & -a_{32}a_{11} & a_{11}a_{22} \end{bmatrix} where A13 = cofactor of (1,3)th element in A i.e.
\tt A_{13}=\begin{vmatrix}a_{21} & a_{22} \\a_{31} & a_{32} \end{vmatrix}
• If A = diag (λ1, λ,...,λn) then A−1 exists if and only if λi ≠ 0 ∀ i and
A−1 = diag (λ1−12−1,....,λn−1)
Also Am = diag (λ1m2m,....,λnm) if m ∈ N
• If a square matrix A satisfies the equation a0 + a1x + a2x+ ... + arx=0, then A is invertible if a0 ≠ 0 and its inverse is given by A^{-1}= − \frac{1}{a_0}[a_1I+a_2A + ...+a_r\ A^{r-1}]
• If AB = I, then BA = I and B = A−1
• If If AB = CA = I, then B = C