Adjoint and Inverse of a Matrix

  • Adjoint of matrix: The  transpose of the cofactors of the given matrix A is called adjoint of the matrix A, and it is denoted by adj A. i.e.
    A = \begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}
    adj A = \begin{vmatrix}c_{11} & c_{21} & c_{31} \\ c_{12} & c_{22} & c_{32} \\ c_{13} & c_{23} & c_{33} \end{vmatrix}
  • Inverse of the matrix: The inverse of the square matrix A is denoted by A−1 and is defined as
    A^{-1} = \frac{1}{|A|}(adj A)
  • If A = \begin{bmatrix}a & b \\c & d \end{bmatrix} \Rightarrow adj A = \begin{bmatrix}d & -b \\ -c & a \end{bmatrix}
  • If A = \begin{bmatrix}a & b \\c & d \end{bmatrix} \Rightarrow A^{-1} = \frac{1}{ad - bc} \begin{bmatrix}d & -b \\ -c & a \end{bmatrix}
  • If A = \begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \Rightarrow A^{-1} = \begin{bmatrix}\frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\ 0 & 0 & \frac{1}{c} \end{bmatrix}
  • Tips on adjoint and Inverse of a matrix
  • If A is square matrix of order n, then |Adj A| = |A|n−1
  • If A is square matrix of order n, then |Adj (Adj A)| = |A|^{(n - 1)^{2}}
  • If A is square matrix of order n, then |Adj (Adj(Adj A))| = |A|^{(n - 1)^{3}}
  • If A is the square matrix of order n, then |A (adj A)| = |A|n
  • If A is the square matrix, then A·(Adj A) = (Adj A)· A = |A| I.
  • If A is the square matrix, then Adj (AT) = (adj A)T
  • For any scalar k and A is the square, matrix then adj (kA) = kn−1 (adj A) where n is the order of matrix.
  • If A and B are square matrices of same order then adj (AB) = (adj B) (adj A).
  • If I is the identify matrix, then I−1 = I
  • If A and B are two invertible matrices, then (AB)−1 = B−1 A−1
  • If A is invertible matrix, then |A^{-1} |= |A|^{-1} = \frac{1}{|A|}
  • If A is the non singular matrix of order n, then Adj(Adj A) = |A|n−2 A
  • If A is the square matrix and is invertible then (AT)−1 = (A−1)T
  • If A1, A2, A3 ........ An are the invertible matrices then \tt (A_{1} \cdot A_{2} ...... A_{n})^{-1} = A_{n}^{-1} \cdot A_{n-1}^{-1}...... A_{2}^{-1} \cdot A_{1}^{-1}
  • If A is the invertible matrix, then \tt (adj A)^{-1} = \frac{A}{|A|} = adj (A^{-1})
  • The inverse of the symmetric matrix is symmetric.
  • The inverse of the diagonal matrix is diagonal matrix.
  • If |A| = 0 ⇒ |adj A| = 0
  • If A is symmetric then adj A is also symmetric.
  • If A is singular matrix then A·(adj A) = (adj A) A = 0

View the Topic in this video From 01:20 To 58:48

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1. Properties of adjoint matrix : If A, B are square matrices of order n and In is corresponding unit matrix, then

    (i) A(adj A) = |A| In = (adj A)A  (Thus A (adj A) is always a scalar matrix)

    (ii) |adj A| = |A|n−1

    (iii) adj (adj A) = |A|n−2 A; |A| ≠ 0.

    (iv) |adj (adj A)| = |A|^{(n-1)^2}

    (v) adj (AT) = (adj A)T

    (vi) adj (AB) = (adj B)(adj A)

    (vii) adj(Am) = (adj A)m, m ∈ N

    (viii) adj(kA) = kn−1(adj A), k ∈ R

     (ix) adj (In) = In

     (x) adj(O) = O

2. The inverse of A is given by A^{-1}=\frac{1}{|A|} . adj A.

3. Properties of inverse matrix: If A and B are invertible matrices of the same order, then

    (i) (A−1)−1 = A

    (ii) (AT)−1 = (A−1)T

    (iii) (AB)−1 = B−1A−1

    (iv) (Ak)−1 = (A−1)k, k ∈ N          [In particular (A2)−1 = (A−1)2]

    (v) adj(A−1) = (adj A)−1

    (vi) |A^{-1}|=\frac{1}{|A|}=|A|^{-1}