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Minors and Co-factors

**Minors:**The minor of any element is the determinant obtained by deleting the row and the column in which that element is lying. Let

**Δ = | a**be a determinant of order

_{ij}| ; i, j = 1, 2, 3**3 × 3**, where

**a**

_{ij}is the element lying in the

**i**

^{th}row and

**j**

^{th}column in the determinant

**Δ**. Then the minor of the element

**a**

_{ij}in the determinant

**Δ**is another determinant of order

**2 × 2**obtained from

**Δ**by deleting the

**i**

^{th}row and

**j**

^{th}column (in which element

**a**

_{ij}lies) and is denoted by

**M**.

_{ij}, i, j = 1, 2, 3**Illustration:**

be a given determinant.

The minors of

**a**

_{11}, a_{12}, a_{13}in the determinant

**Δ = | a**are

_{ij}|, i, j = 1, 2, 3**M**

_{11}, M_{12}, M_{13}respectively, where

The minors of

**a**

_{21}, a_{22}, a_{23}are

**M**

_{21}, M_{22}and

**M**

_{23}respectively, where

The minors of

**a**are

_{31}, a_{32}, a_{33}**M**

_{31}, M_{32}and

**M**

_{33}respectively, where

**Co-factor:**The co-factor of an element

**a**

_{ij}in a determinant

**Δ**is the minor of

**a**

_{ij}with proper signs and is usually denoted by the corresponding capital letter, i.e. the co-factor of

**a**is denoted by

_{ij}**A**

_{ij}.

In other words

**A**

_{ij}= (–1)^{1+j}M_{ij}= (–1)^{1+j}. Minor of a_{ij}in ΔNow the co-factor of

**a**

_{21}in the above illustration is

Similarly,

**Minors and co-factors of the elements in a determinant are equal in magnitude but they differ in sign only.**

Remark:

Remark:

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