## Determinants

# 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 (A
^{T}) = (adj A)^{T} - For any scalar k and A is the square, matrix then adj (kA) = k
^{n−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 (A
^{T})^{−1}= (A^{−1})^{T} - If A
_{1}, A_{2}, A_{3}........ 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 I_{n} is corresponding unit matrix, then

(i) A(adj A) = |A| I_{n} = (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 (A^{T}) = (adj A)^{T}

(vi) adj (AB) = (adj B)(adj A)

(vii) adj(A^{m}) = (adj A)^{m}, m ∈ N

(viii) adj(kA) = k^{n−1}(adj A), k ∈ R

(ix) adj (I_{n}) = I_{n}

(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) (A^{T})^{−1} = (A^{−1})^{T}

(iii) (AB)^{−1} = B^{−1}A^{−1}

(iv) (A^{k})^{−1} = (A^{−1})^{k}, k ∈ N [In particular (A^{2})^{−1} = (A^{−1})^{2}]

(v) adj(A^{−1}) = (adj A)^{−1}

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