In minor of 2 in (2,3) place in [latex]

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  latex a_{ij} /latex, we mean the value of the determinant obtained by deleting the latex i^{th} /latex row and latex j^{th} /latex column of A matrix. It is denoted by latex M_{ij} /latex. 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:Hide latex i^{th} /latex row and latex j^{th} /latex 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.Evaluate the value of the determinant of the matrix made after hiding a row and a column from Step 1.Minor of order 3*3 matrixExample: Consider the square matrix latex A=egin {bmatrix} 2 &-1 & 3 \ 0&4&2 \ 1 & -1& -2 end {bmatrix} /latexSolution: 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 latex egin {bmatrix} 4 &2 \ -1&-2 end {bmatrix} /latex = (4*-2) – (2*-1) = -8+2= -6 = latex M_{11} /latexSince -1 is (1,2) element, we delete first row and second column. The determinant of remaining array latex egin {bmatrix} 0 &2 \ 1&-2 end {bmatrix} /latex = 0*-2-(2*1) = -2 = latex M_{12} /latexThe minor of 3 is latex egin {bmatrix} 0 &4 \ 1&-1 end {bmatrix} /latex = 0-4 = -4 =  latex M_{13} /latexThe minor of 0 is latex egin {bmatrix} -1 &3 \ -1&-2 end {bmatrix} /latex = (-1)(-2)-(3)(-1) = 2+3 = 5 = latex M_{21} /latexThe minor of 4 is latex egin {bmatrix} 2 &3 \ 1&-2 end {bmatrix} /latex = (2)(-2)-(3)(1) = -4-3 = -7 latex M_{22} /latexThe minor of 2 in (2,3) place in latex egin {bmatrix} 2 &-1 \ 1&-1 end {bmatrix} /latex = (2)(-1) – (1)(1) = -2+1 = -1 =  latex M_{23} /latexThe minor of 1 is latex egin {bmatrix} -1&3 \ 4&2 end {bmatrix} /latex = (-1)(2) – (3)(4) = -2-12 = -14 =  latex M_{31} /latexThe minor of (-1) is latex egin {bmatrix} 2&3 \ 0&2 end {bmatrix} /latex = (4)-0 = 4 = latex M_{32} /latexThe minor of (-2) is latex egin {bmatrix} 2&-1 \ 0&4 end {bmatrix} /latex = (2)(4)-0 = 8  =  latex M_{33} /latexMinor of order 2*2 matrixFor a 2*2 matrix, calculation of minors is very simple. Consider the matrix latex P=egin {bmatrix} 2 &6 \ -4&7 end {bmatrix} /latex. For finding minor of 2 we delete first row and first column.For example, latex egin {bmatrix} -2 &6 \ -4&7 end {bmatrix} /latex. So that remaining array is |7| = 7 =  latex M_{11} /latexSimilarly, minors of 6, -4 and 7 will be -4,6,2 respectively.ExerciseFind the minor of the matrix latex A=egin {bmatrix} -1 &-2 &-2 \ 2&1&-2 \ 2&-2&1 end {bmatrix} /latex.Find the minor of matrix latex F=egin {bmatrix} 4 &2 &3 \ 4&0&1 \ 1&1&0 end {bmatrix} /latex.Find the minor of matrixlatex A=egin {bmatrix} 8 &-9 \ -5&6 end {bmatrix} /latex.Find the minor of matrix latex D=egin {bmatrix} 4 &-5 \ 2&1 end {bmatrix} /latex.Find the minor of the matrix latex G=egin {bmatrix} 3 &-4 &1 \ -3&6&-1 \ 4&-6&2 end {bmatrix} /latex.