Find the value of determinant a h g h b f g f c

To evaluate: a h g h b f g f c Expanding along the first row, we get a h g h b f g f c = a b f f c - h h f g c + g h b g f = a b c - f 2 - h h c - f g + g h f - b g = a b c - a f 2 - h 2 c + h f g + g h f - b g 2 = a b c + 2 f g h - a f 2 - b g 2 - c h 2 Hence, the value of given determinant is a b c + 2 f g h - a f 2 - b g 2 - c h 2 .

Find the value of determinant a h g h b f g f c

To evaluate: a h g h b f g f c Expanding along the first row, we get a h g h b f g f c = a b f f c - h h f g c + g h b g f = a b c - f 2 - h h c - f g + g h f - b g = a b c - a f 2 - h 2 c + h f g + g h f - b g 2 = a b c + 2 f g h - a f 2 - b g 2 - c h 2 Hence, the value of given determinant is a b c + 2 f g h - a f 2 - b g 2 - c h 2 .

Find the value of x if x 2 - x + 1 x + 1 x + 1 x + 1 = 0 .

Given, x 2 - x + 1 x + 1 x + 1 x + 1 = 0 ∴ x 2 - x + 1 x + 1 - x + 1 x + 1 = 0 ⇒ x + 1 x 2 - x + 1 - x - 1 = 0 ⇒ x + 1 x 2 - 2 x = 0 ⇒ x x + 1 x - 2 = 0 ⇒ x = 0 or x + 1 = 0 or x - 2 = 0 ⇒ x = 0 , x = - 1 , x = 2 Hence, x = 0 , x = - 1 , x = 2

Find x and y if 4 i i 3 2 i 1 3 i 2 4 5 - 3 i = x + i y , where i 2 = - 1 .

Given, 4 i i 3 2 i 1 3 i 2 4 5 - 3 i = x + i y ∴ 4 i 3 i 2 4 - 3 i - i 3 1 4 5 i + 2 i 1 3 i 2 5 - 3 = x + i y ⇒ 4 i 3 i 3 + 12 - i 3 i - 20 + 2 i - 3 - 15 i 2 = x + i y ⇒ 12 i 4 + 48 i - i 4 + 20 i 3 - 6 i - 30 i 3 = x + i y ⇒ 12 + 48 i - 1 - 20 i - 6 i + 30 i = x + i y ∵ i 2 = - 1 ∴ i 3 = i 2 · i = - i , i 4 = i 2 2 = 1 ⇒ 11 + 52 i = x + i y Now, comparing real part and imaginary part will give x = 11 , y = 52 Hence, the value of x and y is 11 and 52 respectively.

Without expanding evaluate the following determinant. 1 a b + c 1 b c + a 1 c a + b

Let D = 1 a b + c 1 b c + a 1 c a + b Applying C 3 → C 2 + C 3 , we have D = 1 a a + b + c 1 b a + b + c 1 c a + b + c Now, taking a + b + c common from C 3 , we get D = a + b + c 1 a 1 1 b 1 1 c 1 Since, C 1 and C 3 are same, ∴ D = a + b + c × 0 = 0 Hence, the value of given determinant is 0 .

E M B I B E

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 1

EASY

11th Maharashtra Board

IMPORTANT

Earn 100

Find the value of determinant $|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}|$

Important Questions on Determinants and Matrices

EASY

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 2

Find the value of

x

if

|\begin{matrix} x^{2} - x + 1 & x + 1 \\ x + 1 & x + 1 \end{matrix}| = 0

.

EASY

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 2

Find the value of $x$ if $|\begin{array}{r} x & - 1 & 2 \\ 2 x & 1 & - 3 \\ 3 & - 4 & 5 \end{array}| = 29$

MEDIUM

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 3

Find

x

and

y

if

|\begin{matrix} 4 i & i^{3} & 2 i \\ 1 & 3 i^{2} & 4 \\ 5 & - 3 & i \end{matrix}| = x + i y

, where

i^{2} = - 1

.

MEDIUM

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 4

Find the minor and cofactor of element of the determinant

$D = |\begin{array}{r} 2 & - 1 & 3 \\ 1 & 2 & - 1 \\ 5 & 7 & 2 \end{array}|$

HARD

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 5

Evaluate $A = |\begin{matrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{matrix}|$ , also find minor and cofactor of elements in $2^{nd}$ row of determinant and verify $- a_{21} M_{21} + a_{22} M_{22} - a_{23} M_{23} =$ value of $A$ , where $M_{21}, M_{22}, M_{23}$ are minor of $a_{21}, a_{22}, a_{23}$ .

HARD

11th Maharashtra Board

IMPORTANT

Mathematics and Statistics (Arts & Science) Part 1>Chapter 4 - Determinants and Matrices>EXERCISE 4.1>Q 5

Evaluate $A = |\begin{matrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{matrix}|$ , also find minor and cofactor of elements in $2^{nd}$ row of determinant and verify $a_{21} C_{21} + a_{22} C_{22} + a_{23} C_{23} =$ value of $A$ , where $C_{21}, C_{22}, C_{23}$ are cofactor of $a_{21}, a_{22}, a_{23}$ .