Minors
Calculating minors
- Each element of a 3 × 3 matrix has its own minor.
- Minors are denoted as Mij.
- ‘i’ is the row number and ‘j’ is the column number.
- A minor is a second-order determinant (Or determinant of a 2 × 2 matrix) obtained by eliminating the row and the column of the respective element.
- Consider the following matrix ‘A’.
\[\Large
A =
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\]
- In that matrix, every element (a11, a12, etc…) has its own minor.
- The minor of the element a11 (Minor M11) can be obtained by eliminating the row and the column of the element a11, which is row 1 and column 1.
- To compute the minor of the element a11, eliminate its row and column (As shown in the above image), and calculate determinant of the remaining 2 × 2 matrix.
\[\Large
M_{11} =
det
\left\vert
\begin{array}{ccc}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}
\right\vert
\]
\[\Large M_{11} = det~ \left\vert \begin{array}{ccc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right\vert = a_{22}a_{33}~-~a_{23}a_{32} \]
\[\Large M_{11} = det~ \left\vert \begin{array}{ccc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right\vert = a_{22}a_{33}~-~a_{23}a_{32} \]
Also we can calculate the minors of other elements too.
- Let’s calculate the minor of the element a13 (Minor M13)
\[\Large
M_{13} =
det
\left\vert
\begin{array}{ccc}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}
\right\vert
\]
\[\Large M_{13} = det~ \left\vert \begin{array}{ccc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right\vert = a_{21}a_{32}~-~a_{22}a_{31} \]
\[\Large M_{13} = det~ \left\vert \begin{array}{ccc} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right\vert = a_{21}a_{32}~-~a_{22}a_{31} \]
Example
Let’s find the minor of the element ‘1’. (Minor M21)
\[\Large
M_{21} =
det~
\left\vert
\begin{array}{ccc}
1 & 4 \\
2 & 2
\end{array}
\right\vert
= 1×2~-~4×2 = \mathbf{-6}
\]
Which elements minor is shown below?
