## Inverse of a Matrix using Minors, Cofactors and Adjugate

### Minors and Cofactors

**Minor of an element:**

If we take the element of the determinant and delete (remove) the row and column containing that element, the determinant left is called the minor of that element. It is denoted by M_{ij}.

Similarly, we can find the minors of other elements. Using this concept the value of determinant can be

∆ = a_{11}M_{11} – a_{12}M_{12} + a_{13}M_{13}

or, ∆ = – a_{21}M_{21} + a_{22}M_{22} – a_{23}M_{23}

or, ∆ = a_{31}M_{31} – a_{32}M_{32} + a_{33}M_{33}

**Cofactor of an element: **The cofactor of an element a

_{ij}(i.e. the element in the i

^{th}row and j

^{th}column) is defined as (–1)

^{i+j}times the minor of that element. It is denoted by C

_{ij }or A

_{ij or }F

_{ij}.

C

_{ij }= (–1)

^{i+j}M

_{ij}.

where C

_{11 }= (–1)

^{1+1}M

_{11}= +M

_{11}, C

_{12 }= (–1)

^{1+2}M

_{12}= –M

_{12}and C

_{13 }= (–1)

^{1+3}M

_{13}= +M

_{13}

Similarly, we can find the cofactors of other elements.

### Adjugate (also called Adjoint) of a Square Matrix

Let A = [a_{ij}] be a square matrix of order n and let C_{ij }be cofactor a_{ij} of in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A

Thus, adj A = [C_{ij}]^{T} ⇒ (adj A)_{ij} = C_{ji} = cofactor of a_{ij} in A.

**Properties of adjoint matrix:**

If A, B are square matrices of order n and is corresponding unit matrix, then

- A(adj A)=|A|, I
_{n}=(adj A)A

(Thus A (adj A) is always a scalar matrix) - |adj A|= |A|
^{n-1} - adj(adj A) = |A|
^{n-2}A - adj(adj A) =|A|
^{(n-1)^2} - adj(A
^{T}) = (adj A)^{T} - adj(AB) = (adj B)(adj A)
- adj(A
^{m}) = (adj A)^{m}, m ∈ N - adj(kA) = k
^{n-1}(adj A), k ∈ R - adj(I
_{n}) = I_{n} - adj(0) = 0
- A is symmetric ⇒ adj A is also symmetric.
- A is diagonal ⇒ adj A is also diagonal.
- A is triangular ⇒ adj A is also triangular.
- A is singular ⇒ |adj A|= 0

### Adjoint of a Square Matrix Problems with Solutions

**1.**

**Solution:**

**2.**

**Solution:**

### Inverse of a Matrix

A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I_{n} =BA .

In such a case, we say that the inverse of A is B and we write A^{-1} = B. The inverse of A is given by

The necessary and sufficient condition for the existence of the inverse of a square matrix A is that |A| ≠ 0.

**Properties of inverse matrix:**

If A and B are invertible matrices of the same order, then

- (A
^{-1})^{-1}= A - (A
^{T})^{ -1}=(A^{-1})^{T} - (AB)
^{ -1}= B^{-1}A^{-1} - (A
^{k})^{-1 }= (A^{-1})^{k}, k ∈ N [In particular (A^{2})^{-1}=(A^{-1})^{2}] - adj(A
^{-1}) = (adj A)^{-1} - A = diag (a
_{1}a_{2}…a_{n}) ⇒ A^{-1}= diag(a_{1}^{-1}a_{2}^{-1}…a_{n}^{-1}) - A is symmetric ⇒ A
^{-1}is also symmetric. - A is diagonal, |A| ≠ 0 ⇒ A
^{-1}is also diagonal. - A is a scalar matrix ⇒ A
^{-1}is also a scalar matrix. - A is triangular, |A| ≠ 0 ⇒ A
^{-1 }is also triangular. - Every invertible matrix possesses a unique inverse.
- Cancellation law with respect to multiplication.

If A is a non-singular matrix i.e., if |A|≠ 0 ,then A^{-1}exists and AB = AC ⇒ A^{-1}(AB) = A^{-1}(AC)

⇒ (A^{-1}A)B =(A^{-1}A)C

⇒ IB = IC ⇒ B=C

∴ AB=AC ⇒ B = C ⇔|A|≠ 0.

### Inverse of a Matrix Problems with Solutions

**1.**

**Solution:**

**2.**

**Solution:**

**3.**

**Solution:**