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Inverse of a Matrix

Write the min...

Question

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

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

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

(iv) $A = [\begin{matrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{matrix}]$

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

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

(vii) $A = [\begin{matrix} 2 & - 1 & 0 & 1 \\ - 3 & 0 & 1 & - 2 \\ 1 & 1 & - 1 & 1 \\ 2 & - 1 & 5 & 0 \end{matrix}]$

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Solution

(i)
$M_{11} = - 1 M_{21} = 20 C_{i j} = {(- 1)}^{i + j} M_{i j} C_{11} = {(- 1)}^{1 + 1} (- 1) = - 1 C_{21 =} {(- 1)}^{1 + 2} (20) = - 20 D = (- 1 \times 5) - (20 \times 0) = - 5$

(ii)
$M_{11} = 3 M_{21} = 4 C_{i j} = {(- 1)}^{i + j} M_{i j} C_{11} = {(- 1)}^{1 + 1} M_{11} = 3 C_{21} = {(- 1)}^{2 + 1} M_{21} = - (4) = - 4 D = (3 \times - 1) - (4 \times 2) = - 3 - 8 = - 11$

(iii)
$M_{11} = |\begin{matrix} - 1 & 2 \\ 5 & 2 \end{matrix}| = - 2 - 10 = - 12 M_{21 =} |\begin{matrix} - 3 & 2 \\ 5 & 2 \end{matrix}| = - 6 - 10 = - 16 M_{31 =} |\begin{matrix} - 3 & 2 \\ - 1 & 2 \end{matrix}| = - 6 + 2 = - 4 C_{11} = {(- 1)}^{1 + 1} M_{11} = - 12 C_{21 =} {(- 1)}^{2 + 1} M_{21} = - (- 16) = 16 C_{31} = {(- 1)}^{3 + 1} M_{31} = - 4 D = 1 (- 12) + 3 (8 - 6) + 2 (20 + 3) = - 12 + 6 + 46 = 40$

(iv)
$M_{11} = |\begin{matrix} b & c a \\ c & a b \end{matrix}| = a b^{2} - c^{2} a = a (b^{2} - c^{2}) M_{21} = |\begin{matrix} a & b c \\ c & a b \end{matrix}| = a^{2} b - c^{2} b = b (a^{2} - c^{2}) M_{31 =} |\begin{matrix} a & b c \\ b & c a \end{matrix}| = a^{2} c - b^{2} c = c (a^{2} - b^{2}) C_{11} = {(- 1)}^{1 + 1} M_{11} = a (b^{2} - c^{2}) C_{21} = {(- 1)}^{2 + 1} M_{21} = - b (a^{2} - c^{2}) C_{31} = {(- 1)}^{3 + 1} M_{31} = c (a^{2} - b^{2}) D = 1 . a (b^{2} - c^{2}) - a (a b - c a) + b . c (c - b) = a b^{2} - a c^{2} - a^{2} b + a^{2} c + c^{2} b - b^{2} c = a^{2} (c - b) + b^{2} (a - c) + c^{2} (b - a)$

(v)
$M_{11} = |\begin{matrix} 5 & 0 \\ 7 & 1 \end{matrix}| = 5 - 0 = 5 M_{21} = |\begin{matrix} 2 & 6 \\ 7 & 1 \end{matrix}| = 2 - 42 = - 40 M_{31} = |\begin{matrix} 2 & 6 \\ 5 & 0 \end{matrix}| = 0 - 30 = - 30 C_{11} = {(- 1)}^{1 + 1} M_{11} = 5 C_{21} = {(- 1)}^{2 + 1} M_{21} = - (- 40) C_{31} = {(- 1)}^{3 + 1} M_{31} = - 30 D = 0 (5 - 0) - 2 (1 - 0) + 6 (7 - 15) = - 2 - 48 = - 50$

(vi)
$M_{11} = |\begin{matrix} b & f \\ f & c \end{matrix}| = b c - f^{2} M_{21 =} |\begin{matrix} h & g \\ f & c \end{matrix}| = h c - f g M_{31 =} |\begin{matrix} h & g \\ b & f \end{matrix}| = h f - g b C_{11} = {(- 1)}^{1 + 1} M_{11} = b c - f^{2} C_{21} = {(- 1)}^{2 + 1} M_{11} = - (h c - f g) = f g - h c C_{31} = {(- 1)}^{3 + 1} M_{11} = h f - g b D = a (b c - f^{2}) - h (h c - f g) + g (f h - b g) = a b c - a f^{2} - h^{2} c + f g h + f g h - b g^{2} = a b c + 2 h f g - a f^{2} - b g^{2} - c h^{2}$

(vii)

$M_{11} = 0 (0 - 5) - 1 (0 + 1) - 2 (5 - 1) = - 1 - 8 = - 9 M_{21} = - 1 (0 - 5) + 1 (5 - 1) = 5 + 4 = 9 M_{31} = - 1 (0 + 10) + 1 (0 + 1) = - 10 + 1 = - 9 M_{41} = - 1 (1 - 2) + 1 (0 - 1) = 1 - 1 = 0 C_{11} = {(- 1)}^{1 + 1} M_{11} = - 9 C_{21} = {(- 1)}^{2 + 1} M_{21} = (- 1) \times 9 C_{31} = {(- 1)}^{3 + 1} M_{31} = - 9 C_{41} = {(- 1)}^{4 + 1} M_{41} = 0 D = 2 |\begin{matrix} 0 & 1 & - 2 \\ 1 & - 1 & 1 \\ - 1 & 5 & 0 \end{matrix}| + 1 |\begin{matrix} - 3 & 1 & - 2 \\ 1 & - 1 & 1 \\ 2 & 5 & 0 \end{matrix}| - 1 |\begin{matrix} - 3 & 0 & 1 \\ 1 & 1 & - 1 \\ 2 & - 1 & 5 \end{matrix}| = - 18 - 27 + 15 = 30$

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