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Minor of Matrices

In a square matrix, each element possesses its own minor. The minor is defined as a value obtained from the determinant of a square matrix by deleting out a row and a column corresponding to the element of a matrix.

Given a square matrix A, by minor of an element   [a_{ij}] , we mean the value of the determinant obtained by deleting the  i^{th} row and  j^{th} column of A matrix. It is denoted by  M_{ij} . In order to find the minor of the square matrix, we have to erase out a row & a column one by one at the time & calculate their determinant, until all the minors are computed. The following are the steps to calculate minor from a matrix:

  1. Hide  i^{th} row and  j^{th} column one by one from given matrix, where i refer to m and j refers to n that is the total number of rows and columns in matrices.
  2. Evaluate the value of the determinant of the matrix made after hiding a row and a column from Step 1.

Minor of 3×3 Matrix

Consider the 3*3 matrix  A=\begin {bmatrix} a &b &c \\ d&e&f \\ g&h&i \end {bmatrix}  We had to hide the first row and column in order to find the minors of matrices.

M_{11}=\begin {bmatrix} e&f \\ h&i \end {bmatrix} = ei – hf

M_{12}=\begin {bmatrix} d&f \\ g&i \end {bmatrix} = di – fg

M_{13}=\begin {bmatrix} d&e \\ g&h \end {bmatrix} = dh – eg

M_{21}=\begin {bmatrix} b&c \\ h&i \end {bmatrix} = bi – ch

M_{22}=\begin {bmatrix} a&c \\ g&i \end {bmatrix} = ai – cg

M_{23}=\begin {bmatrix} a&b \\ g&h \end {bmatrix} = ah – bg

M_{31}=\begin {bmatrix} b&c \\ e&f \end {bmatrix} = bf – ce

M_{32}=\begin {bmatrix} a&c \\ d&f \end {bmatrix} = af – cd

M_{33}=\begin {bmatrix} a&b \\ d&e \end {bmatrix} = ae – bd

Example: Consider the 3*3 matrix  A=\begin {bmatrix} 2 &-1 & 3 \\ 0&4&2 \\ 1 & -1& -2 \end {bmatrix}

Solution: We first calculate minor of element 2. Since it is (1,1) element of A, we delete first row and first column, so that determinant of remaining array is  \begin {bmatrix} 4 &2 \\ -1&-2 \end {bmatrix} = (4*-2) – (2*-1) = -8+2= -6 =  M_{11}

Since -1 is (1,2) element, we delete first row and second column. The determinant of remaining array  \begin {bmatrix} 0 &2 \\ 1&-2 \end {bmatrix} = 0*-2-(2*1) = -2 =  M_{12}

The minor of 3 is  \begin {bmatrix} 0 &4 \\ 1&-1 \end {bmatrix} = 0-4 = -4 =   M_{13}

The minor of 0 is  \begin {bmatrix} -1 &3 \\ -1&-2 \end {bmatrix} = (-1)(-2)-(3)(-1) = 2+3 = 5 =  M_{21}

The minor of 4 is  \begin {bmatrix} 2 &3 \\ 1&-2 \end {bmatrix} = (2)(-2)-(3)(1) = -4-3 = -7 M_{22}

The minor of 2 in (2,3) place in  \begin {bmatrix} 2 &-1 \\ 1&-1 \end {bmatrix} = (2)(-1) – (1)(1) = -2+1 = -1 =  M_{23}

The minor of 1 is  \begin {bmatrix} -1&3 \\ 4&2 \end {bmatrix} = (-1)(2) – (3)(4) = -2-12 = -14 =  M_{31}

The minor of (-1) is  \begin {bmatrix} 2&3 \\ 0&2 \end {bmatrix} = (4)-0 = 4 = M_{32}

The minor of (-2) is  \begin {bmatrix} 2&-1 \\ 0&4 \end {bmatrix} = (2)(4)-0 = 8  =  M_{33}

Minor of 2×2 Matrix

For a 2*2 matrix, calculation of minors is very simple. Let us consider a 2 x 2 matrix  A=\begin {bmatrix} a&b \\ c&d \end {bmatrix} . We had to hide the first row and column to find the minors of matrices.

M_{11}=\begin {bmatrix} -&- \\ -&d \end {bmatrix} = d

M_{12}=\begin {bmatrix} -&- \\ c&- \end {bmatrix} = c

M_{21}=\begin {bmatrix} -&b \\ -&- \end {bmatrix} = b

M_{22}=\begin {bmatrix} a&- \\ -&- \end {bmatrix} = a

Example: Consider the matrix  P=\begin {bmatrix} 2 &6 \\ -4&7 \end {bmatrix} . For finding minor of 2 we delete first row and first column.

Solution: \begin {bmatrix} -2 &6 \\ -4&7 \end {bmatrix} . So that remaining array is |7| = 7 =  M_{11}

Similarly, minors of 6, -4 and 7 will be -4,6,2 respectively.

Exercise

  1. Find the minor of the matrix A=\begin {bmatrix} -1 &-2 &-2 \\ 2&1&-2 \\ 2&-2&1 \end {bmatrix} .
  2. Find the minor of matrix F=\begin {bmatrix} 4 &2 &3 \\ 4&0&1 \\ 1&1&0 \end {bmatrix} .
  3. Find the minor of matrix A=\begin {bmatrix} 8 &-9 \\ -5&6 \end {bmatrix} .
  4. Find the minor of matrix D=\begin {bmatrix} 4 &-5 \\ 2&1 \end {bmatrix} .
  5. Find the minor of the matrix G=\begin {bmatrix} 3 &-4 &1 \\ -3&6&-1 \\ 4&-6&2 \end {bmatrix} .


Comments

  2. sean says

    second the first comment, major error that could negatively affect students’ understanding of minors. please change or delete this page to prevent further damage.

    Reply

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