Exercise 4.3 Practice
Minors and Cofactors: Calculation and Application
Q1
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Write Minors and Cofactors of the elements of the following determinants:
(i) \( \begin{vmatrix} 3 & -4 \\ 2 & 5 \end{vmatrix} \)
(ii) \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
(i) \( \begin{vmatrix} 3 & -4 \\ 2 & 5 \end{vmatrix} \)
(ii) \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
Part (i) Solution
Minors:
\( M_{11} = 5 \), \( M_{12} = 2 \)
\( M_{21} = -4 \), \( M_{22} = 3 \)
Cofactors (\( A_{ij} = (-1)^{i+j}M_{ij} \)):
\( A_{11} = 5 \), \( A_{12} = -2 \)
\( A_{21} = 4 \), \( A_{22} = 3 \)
\( M_{11} = 5 \), \( M_{12} = 2 \)
\( M_{21} = -4 \), \( M_{22} = 3 \)
Cofactors (\( A_{ij} = (-1)^{i+j}M_{ij} \)):
\( A_{11} = 5 \), \( A_{12} = -2 \)
\( A_{21} = 4 \), \( A_{22} = 3 \)
Part (ii) Solution
Minors: \( M_{11} = d, M_{12} = c, M_{21} = b, M_{22} = a \)
Cofactors: \( A_{11} = d, A_{12} = -c, A_{21} = -b, A_{22} = a \)
Cofactors: \( A_{11} = d, A_{12} = -c, A_{21} = -b, A_{22} = a \)
Q2
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Write Minors and Cofactors of the elements of the following determinant:
\( \Delta = \begin{vmatrix} 1 & 0 & 2 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix} \)
\( \Delta = \begin{vmatrix} 1 & 0 & 2 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix} \)
Minors
\( M_{11} = 5(2) - (-1)(1) = 11 \)
\( M_{12} = 3(2) - (-1)(0) = 6 \)
\( M_{13} = 3(1) - 5(0) = 3 \)
\( M_{21} = 0(2) - 2(1) = -2 \)
\( M_{22} = 1(2) - 2(0) = 2 \)
\( M_{23} = 1(1) - 0(0) = 1 \)
\( M_{31} = 0(-1) - 2(5) = -10 \)
\( M_{32} = 1(-1) - 2(3) = -7 \)
\( M_{33} = 1(5) - 0(3) = 5 \)
\( M_{12} = 3(2) - (-1)(0) = 6 \)
\( M_{13} = 3(1) - 5(0) = 3 \)
\( M_{21} = 0(2) - 2(1) = -2 \)
\( M_{22} = 1(2) - 2(0) = 2 \)
\( M_{23} = 1(1) - 0(0) = 1 \)
\( M_{31} = 0(-1) - 2(5) = -10 \)
\( M_{32} = 1(-1) - 2(3) = -7 \)
\( M_{33} = 1(5) - 0(3) = 5 \)
Cofactors
\( A_{11}=11, A_{12}=-6, A_{13}=3 \)
\( A_{21}=2, A_{22}=2, A_{23}=-1 \)
\( A_{31}=-10, A_{32}=7, A_{33}=5 \)
\( A_{21}=2, A_{22}=2, A_{23}=-1 \)
\( A_{31}=-10, A_{32}=7, A_{33}=5 \)
Q3
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Using Cofactors of elements of the second row, evaluate the determinant:
\( \Delta = \begin{vmatrix} 2 & 5 & 3 \\ 1 & 0 & 4 \\ 3 & 2 & 1 \end{vmatrix} \)
\( \Delta = \begin{vmatrix} 2 & 5 & 3 \\ 1 & 0 & 4 \\ 3 & 2 & 1 \end{vmatrix} \)
Identify Elements
Elements of R2 are \( a_{21}=1, a_{22}=0, a_{23}=4 \).
Calculate Cofactors
\( A_{21} = (-1)^{2+1} \begin{vmatrix} 5 & 3 \\ 2 & 1 \end{vmatrix} = -(5-6) = 1 \)
\( A_{22} = (-1)^{2+2} \begin{vmatrix} 2 & 3 \\ 3 & 1 \end{vmatrix} = (2-9) = -7 \)
\( A_{23} = (-1)^{2+3} \begin{vmatrix} 2 & 5 \\ 3 & 2 \end{vmatrix} = -(4-15) = 11 \)
\( A_{22} = (-1)^{2+2} \begin{vmatrix} 2 & 3 \\ 3 & 1 \end{vmatrix} = (2-9) = -7 \)
\( A_{23} = (-1)^{2+3} \begin{vmatrix} 2 & 5 \\ 3 & 2 \end{vmatrix} = -(4-15) = 11 \)
Evaluate
\( \Delta = a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23} \)
\( = 1(1) + 0(-7) + 4(11) = 1 + 0 + 44 = 45 \).
\( = 1(1) + 0(-7) + 4(11) = 1 + 0 + 44 = 45 \).
Value = 45
Q4
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Using Cofactors of elements of the third column, evaluate the determinant:
\( \Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix} \)
\( \Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix} \)
Cofactors of C3
\( A_{13} = (z-y) \)
\( A_{23} = -(z-x) = (x-z) \)
\( A_{33} = (y-x) \)
\( A_{23} = -(z-x) = (x-z) \)
\( A_{33} = (y-x) \)
Calculate Delta
\( \Delta = yz(z-y) + zx(x-z) + xy(y-x) \)
\( = yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y \)
Factorizing this expression (cyclic order):
\( = (x-y)(y-z)(z-x) \).
\( = yz^2 - y^2z + zx^2 - z^2x + xy^2 - x^2y \)
Factorizing this expression (cyclic order):
\( = (x-y)(y-z)(z-x) \).
Value = \( (x-y)(y-z)(z-x) \)
Q5
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If \( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}
\end{vmatrix} \) and \( A_{ij} \) is cofactor of \( a_{ij} \), then value of \( \Delta \) is given by:
(A) \( a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33} \)
(B) \( a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{31} \)
(C) \( a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13} \)
(D) \( a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31} \)
(A) \( a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33} \)
(B) \( a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{31} \)
(C) \( a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13} \)
(D) \( a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31} \)
Property
The value of a determinant is the sum of the product of elements of any row (or column) with their
corresponding cofactors.
Option (D) represents expansion along the first column.
Correct Option: (D)