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Inequalities of Integrals

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Question

Evaluate the following determinant:
(i) $|\begin{matrix} 1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38 \end{matrix}|$

(ii) $|\begin{matrix} 67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26 \end{matrix}|$

(iii) $|\begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix}|$

(iv) $|\begin{matrix} 1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2 \end{matrix}|$

(v) $|\begin{matrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \end{matrix}|$

(vi) $|\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}|$

(vii) $|\begin{matrix} 1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9 \end{matrix}|$

(viii)

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Solution

$(i) ∆ = |\begin{matrix} 1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38 \end{matrix}| = 1 |\begin{matrix} 6 & 10 \\ 11 & 38 \end{matrix}| - 3 |\begin{matrix} 2 & 10 \\ 31 & 38 \end{matrix}| + 5 |\begin{matrix} 2 & 6 \\ 31 & 11 \end{matrix}| = 1 (228 - 110) - 3 (76 - 310) + 5 (22 - 186) = 1 (118) - 3 (- 234) + 5 (- 164) = 118 + 702 - 820 = 0 (ii) ∆ = |\begin{matrix} 67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26 \end{matrix}| = 67 (338 - 336) - 19 (1014 - 1134) + 21 (936 - 1053) = 67 (2) - 19 (- 120) + 21 (- 117) = 134 + 2280 - 2457 = - 43 (iii) ∆ = |\begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix}| = a |\begin{matrix} b & f \\ f & c \end{matrix}| - h |\begin{matrix} h & f \\ g & c \end{matrix}| + g |\begin{matrix} h & b \\ g & f \end{matrix}| = a (b c - f^{2}) - h (h c - f g) + g (h f - g b) = a b c - a f^{2} - h^{2} c + f g h + f g h - g^{2} b = a b c + 2 f g h - a f^{2} - c h^{2} - b g^{2} (iv) ∆ = |\begin{matrix} 1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2 \end{matrix}| = 1 |\begin{matrix} - 1 & 2 \\ 5 & 2 \end{matrix}| - (- 3) |\begin{matrix} 4 & 2 \\ 3 & 2 \end{matrix}| + 2 |\begin{matrix} 4 & - 1 \\ 3 & 5 \end{matrix}| = 1 (- 2 - 10) + 3 (8 - 6) + 2 (20 + 3) = (- 12) + 6 + 46 = 40 (v) ∆ = |\begin{matrix} 1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25 \end{matrix}| = 1 |\begin{matrix} 9 & 16 \\ 16 & 25 \end{matrix}| - 4 |\begin{matrix} 4 & 16 \\ 9 & 25 \end{matrix}| + 9 |\begin{matrix} 4 & 9 \\ 9 & 16 \end{matrix}| = 1 (225 - 256) - 4 (100 - 144) + 9 (64 - 81) = 1 (- 31) - 4 (- 44) + 9 (- 17) = - 31 + 176 - 153 = - 8 (vi) ∆ = |\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}| = 6 (- 2 - 10) - (- 3) (4 + 20) + 2 (10 - 10) = - 72 + 72 + 0 = - 72 + 72 = 0 (vii) ∆ = |\begin{matrix} 1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9 \end{matrix}| = 1 |\begin{matrix} 9 & 27 & 1 \\ 27 & 1 & 3 \\ 1 & 3 & 9 \end{matrix}| - 3 |\begin{matrix} 3 & 27 & 1 \\ 9 & 1 & 3 \\ 27 & 3 & 9 \end{matrix}| + 9 |\begin{matrix} 3 & 9 & 1 \\ 9 & 27 & 3 \\ 27 & 1 & 9 \end{matrix}| - 27 |\begin{matrix} 3 & 9 & 27 \\ 9 & 27 & 1 \\ 27 & 1 & 3 \end{matrix}| = 1 \{9 (9 - 9) - 27 (243 - 3) + 1 (81 - 1)\} - 3 \{3 (9 - 9) - 27 (81 - 81) + 1 (27 - 27)\} + 9 \{3 (243 - 3) - 9 (81 - 81) + 1 (9 - 729)\} - 27 \{(81 - 1) - 9 (27 - 27) + 27 (9 - 729)\} = 1 \{0 - 6480 + 80\} - 3 \{0 - 0 + 0\} + 9 \{720 - 0 - 720\} - 27 \{80 - 0 - 19440\} = - 6400 + 522720 = 516320$

(viii)

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Similar questions

Q. Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case:
(i)

A = [\begin{matrix} 5 & 20 \\ 0 & - 1 \end{matrix}]

A = [\begin{matrix} - 1 & 4 \\ 2 & 3 \end{matrix}]

A = [\begin{matrix} 1 & - 3 & 2 \\ 4 & - 1 & 2 \\ 3 & 5 & 2 \end{matrix}]

A = [\begin{matrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{matrix}]

A = [\begin{matrix} 0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1 \end{matrix}]

A = [\begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix}]

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Match the following

(1)–(iii), (viii); (2)–(i), (iv); (3)–(ii), (vi); (4)-(ii),(v); (5)–(iii), (iv); (6)–(i) (vii); (7)–(ii), (iv)
(1)–(ii), (vii); (2)–(i), (iv); (3)–(ii), (vi); (4)-(ii),(v); (5)–(ii), (iv); (6)–(i) (viii); (7)–(iii), (iv)
(1)–(ii), (vii); (2)–(i), (vii); (3)–(iii), (iv); (4)-(i),(v); (5)–(ii), (vi); (6)–(i) (vi); (7)–(i), (iv)
(1)–(ii), (vii); (2)–(i), (iv); (3)–(ii), (vi); (4)-(ii),(iv); (5)–(i), (viii); (6)–(iii) (iv); (7)–(iii), (vii)

Evaluate the determinant :

∣ ∣ ∣ ∣ \begin{matrix} 6 & - 3 & 2 2 & - 1 & 2 - 10 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣

Q. Express each of the following as a rational number:

(i)

{(\frac{2}{3})}^{5}

{(\frac{- 8}{5})}^{3}

{(\frac{- 13}{11})}^{2}

{(\frac{1}{6})}^{3}

{(\frac{- 1}{2})}^{5}

{(\frac{- 3}{2})}^{4}

{(\frac{- 4}{7})}^{3}

(viii) (−1)_⁹

Q. Evaluate the following determinant :

∣ ∣ ∣ ∣ \begin{matrix} 1 & - 3 & 2 4 & - 1 & 2 3 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣

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