Using cofactors of elements of second row, evaluate ∆ = 5 3 8 2 0 1 1 2 3 .

We know that, Determinant(Δ) = sum of the product of elements of any row ( or column ) with their corresponding cofactors. Therefore, ∆ = 5 3 8 2 0 1 1 2 3 = a 21 A 21 + a 22 A 22 + a 23 A 23 Here, a 21 = 2 , a 22 = 0 , a 23 = 1 and A 21 = - 1 2 + 1 3 8 2 3 = - 9 - 16 = 7 A 22 = - 1 2 + 2 5 8 1 3 = 15 - 8 = 7 A 23 = - 1 2 + 3 5 3 1 2 = - 10 - 3 = - 7 ∆ = 5 3 8 2 0 1 1 2 3 = 2 7 + 0 7 + 1 - 7 = 7

Using Cofactors of elements of third column, evaluate △ = 1 x y z 1 y z x 1 z x y .

Given that △ = 1 x y z 1 y z x 1 z x y We know that, ∆ = The sum of the product of elements of any row ( or column ) with their corresponding cofactors. Therefore, ∆ = a 13 A 13 + a 23 A 23 + a 33 A 33 a 13 = y z , a 23 = z x , a 33 = x y Calculating cofactors of third column i,e. A 13 , A 23 , and A 33 M 13 = 1 x y 1 y z x 1 z x y = 1 y 1 z = 1 × z - 1 × y = z - y M 23 = 1 x y 1 y zx 1 z x y = 1 x 1 z = 1 × z - 1 × x = z - x M 33 = 1 x yz 1 y zx 1 z xy = 1 x 1 y = 1 × y - 1 × x = y - x Cofactor of a i j = A i j = ( - 1 ) i + j · M i j A 13 = ( - 1 ) 1 + 3 M 13 = ( - 1 ) 4 · ( z - y ) = z - y A 23 = ( - 1 ) 2 + 3 · M 23 = ( - 1 ) 5 · ( z - x ) = ( - 1 ) ( z - x ) = x - z A 33 = ( - 1 ) 3 + 3 · M 33 = ( - 1 ) 6 · M 33 = 1 · ( y - x ) = y - x Now, ∆ = a 13 A 13 + a 23 A 23 + a 33 A 33 = y z ( z - y ) + z x ( x - z ) + x y ( y - x ) = y z 2 - y 2 z + z x 2 - z 2 x + x y 2 - x 2 y = y z 2 - y 2 z + x y 2 - z 2 x + z x 2 - x 2 y = y z ( z - y ) + x y 2 - z 2 + x 2 ( z - y ) = - y z ( y - z ) + x y 2 - z 2 - x 2 ( y - z ) = - y z ( y - z ) + x ( y + z ) ( y - z ) - x 2 ( y - z ) = ( y - z ) - y z + x ( y + z ) - x 2 = ( y - z ) - y z + x y + x z - x 2 = ( y - z ) ( z ( x - y ) + x ( y - x ) ) = ( y - z ) ( z ( x - y ) - x ( x - y ) ) = ( y - z ) ( ( z - x ) ( x - y ) ) = ( x - y ) ( y - z ) ( z - x )

If ∆ = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , A i j is Cofactors of a i j , then value of ∆ is given by

a 11 A 11 + a 21 A 21 + a 31 A 31

a 11 A 31 + a 12 A 32 + a 13 A 33

a 11 A 11 + a 12 A 21 + a 13 A 31

a 21 A 11 + a 22 A 12 + a 23 A 13

E M B I B E

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4

Haryana Board Solutions for Chapter: Determinants, Exercise 4: EXERCISE 4.4

Author:Haryana Board

Haryana Board Mathematics Solutions for Exercise - Haryana Board Solutions for Chapter: Determinants, Exercise 4: EXERCISE 4.4

Attempt the free practice questions on Chapter 4: Determinants, Exercise 4: EXERCISE 4.4 with hints and solutions to strengthen your understanding. MATHEMATICS PART I Textbook for Class XII solutions are prepared by Experienced Embibe Experts.

Questions from Haryana Board Solutions for Chapter: Determinants, Exercise 4: EXERCISE 4.4 with Hints & Solutions

EASY

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 1

Write Minors and Cofactors of the elements of determinant:
$|\begin{matrix} 2 & - 4 \\ 0 & 3 \end{matrix}|$

EASY

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 1

Write Minors and Cofactors of the elements of determinant:
$|\begin{matrix} a & c \\ b & d \end{matrix}|$

EASY

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 2

Write Minors and Cofactors of the elements of determinant:
$|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}|$

EASY

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 2

Write the Minors and Cofactors of the elements of the given determinant.
$|\begin{matrix} 1 & 0 & 4 \\ 3 & 5 & - 1 \\ 0 & 1 & 2 \end{matrix}|$

EASY

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 3

Using cofactors of elements of second row, evaluate $∆ = |\begin{matrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{matrix}| .$

MEDIUM

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 4

Using Cofactors of elements of third column, evaluate $△ = |\begin{array}{l} 1 & x & y z \\ 1 & y & z x \\ 1 & z & x y \end{array}|$ .

EASY

12th Haryana Board

IMPORTANT

MATHEMATICS PART I Textbook for Class XII>Chapter 4 - Determinants>EXERCISE 4.4>Q 5

If $∆ = |\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}|, A_{i j}$ is Cofactors of $a_{i j},$ then value of $∆$ is given by

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