Without expanding, show that the values of each of the following determinants are zero:

Solution:

$(i) \Delta=\left|\begin{array}{ccc}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{array}\right|$

$=\left|\begin{array}{lll}0 & 2 & 7 \\ 0 & 3 & 5 \\ 0 & 4 & 3\end{array}\right| \quad$ [Applying $C_{1} \rightarrow C_{1}-4 C_{2}$ ]

$\Rightarrow \Delta=0$

$(i i) \Delta=\left|\begin{array}{ccc}6 & -3 & 2 \\ 2 & -1 & 2 \\ -10 & 5 & 2\end{array}\right|$

$=\left|\begin{array}{ccc}0 & -3 & 2 \\ 0 & -1 & 2 \\ 0 & 5 & 2\end{array}\right| \quad$ [Applying $C_{1} \rightarrow C_{1}+2 C_{2}$ ]

$\Rightarrow \Delta=0$

$\left({ }_{i i i}\right) \Delta=\left|\begin{array}{ccc}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{array}\right|$

$=\left|\begin{array}{ccc}2 & 3 & 7 \\ 13 & 17 & 5 \\ 13 & 17 & 5\end{array}\right| \quad\left[\right.$ Applying $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$

$\Rightarrow \Delta=0$

$(i v) \Delta=\left|\begin{array}{ccc}\frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & a c \\ \frac{1}{c} & c^{2} & a b\end{array}\right|$

$=\left|\begin{array}{ccc}1 & a^{3} & a b c \\ 1 & b^{3} & a b c \\ 1 & c^{3} & a b c\end{array}\right| \quad\left[\right.$ Applying $R_{1} \rightarrow a R_{1}, R_{2} \rightarrow b R_{2}$ and $\left.R_{3} \rightarrow c R_{3}\right]$

$=a b c\left|\begin{array}{ccc}1 & a^{3} & 1 \\ 1 & b^{3} & 1 \\ 1 & c^{3} & 1\end{array}\right|$

$\Rightarrow \Delta=0$

$(v) \Delta=\left|\begin{array}{ccc}a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\ 4 a+b & 5 a+b & 6 a+b\end{array}\right|$

$=\left|\begin{array}{ccc}a & a & a \\ 2 a & 2 a & 2 a \\ 4 a+b & 5 a+b & 6 a+b\end{array}\right| \quad$ [Applying $R_{1} \rightarrow R_{2}-R_{1}$ and $\left.R_{2} \rightarrow R_{3}-R_{2}\right]$

$=2\left|\begin{array}{ccc}a & a & a \\ a & a & a \\ 4 a+b & 5 a+b & 6 a+b\end{array}\right|=0$

$(v i) \Delta=\left|\begin{array}{lll}1 & a & a^{2}-b c \\ 1 & b & b^{2}-a c \\ 1 & c & c^{2}-a b\end{array}\right|$

$=\left|\begin{array}{ccc}0 & a-b & a^{2}-b c-b^{2}+a c \\ 0 & b-c & b^{2}-a c-c^{2}+a b \\ 1 & c & c^{2}-a b\end{array}\right|$          $\left[\right.$ Applying $\left.R_{1} \rightarrow R_{1}-R_{2}, R_{2} \rightarrow R_{2}-R_{3}\right]$

$=\left|\begin{array}{ccc}0 & a-b & (a-b)(a+b)+c(a-b) \\ 0 & b-c & (b-c)(b+c)+a(b-c) \\ 1 & c & c^{2}-a b\end{array}\right|$

$=(a-b)(b-c)\left|\begin{array}{ccc}0 & 1 & (a+b+c) \\ 0 & 1 & (a+b+c) \\ 1 & c & c^{2}-a b\end{array}\right|$

$\Rightarrow \Delta=0$

$(v i i) \Delta=\left|\begin{array}{lll}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{array}\right|$

$=\left|\begin{array}{lll}1 & 1 & 6 \\ 7 & 7 & 4 \\ 2 & 2 & 3\end{array}\right| \quad\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}-8 C_{3}\right]$

$\Rightarrow \Delta=0$

$($ viii $) \Delta=\left|\begin{array}{ccc}0 & x & y \\ -x & 0 & z \\ -y & -z & 0\end{array}\right|$

$=\frac{x y z}{x y z}\left|\begin{array}{ccc}0 & x & y \\ -x & 0 & z \\ -y & -z & 0\end{array}\right|$

$=\frac{1}{x y z}\left|\begin{array}{ccc}0 & x z & y z \\ -x y & 0 & z y \\ -y x & -z x & 0\end{array}\right|$

$=\frac{1}{x y z}\left|\begin{array}{ccc}-2 x y & 0 & 2 y z \\ -x y & 0 & z y \\ -y x & -z x & 0\end{array}\right| \quad\left[\right.$ Applying $\left.R_{1} \rightarrow R_{1}+R_{2}+R_{3}\right]$

$=\frac{1}{x y z}\left|\begin{array}{ccc}0 & 0 & 0 \\ -x y & 0 & z y \\ -y x & -z x & 0\end{array}\right|=0 \quad$ [Applying $R_{1} \rightarrow R_{1}-2 R_{2}$ ]

$(i x) \Delta=\left|\begin{array}{lll}1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{array}\right|$

$=\left|\begin{array}{lll}1 & 1 & 6 \\ 7 & 7 & 4 \\ 3 & 3 & 2\end{array}\right|=0 \quad\left[\right.$ Applying $\left.C_{2} \rightarrow C_{2}-7 C_{3}\right]$

(x) $\Delta=\left|\begin{array}{llll}1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2}\end{array}\right|$

$=\left|\begin{array}{cccc}1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 9 & 16 & 25 & 36 \\ 16 & 25 & 36 & 49\end{array}\right|$

$=\left|\begin{array}{cccc}1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 5 & 7 & 9 & 11 \\ 7 & 9 & 11 & 13\end{array}\right|$          [Applying $R_{3} \rightarrow R_{3}-R_{2}$ and $R_{4} \rightarrow R_{4}-R_{3}$ ]

$=\left|\begin{array}{cccc}1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 7 & 9 & 11 & 13 \\ 7 & 9 & 11 & 13\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{3} \rightarrow 2+R_{3}\right]$

(xi) $\Delta=\left|\begin{array}{ccc}a & b & c \\ a+2 x & b+2 y & c+2 z \\ x & y & z\end{array}\right|$

$=\left|\begin{array}{lll}1 & 1 & 6 \\ 7 & 7 & 4 \\ 3 & 3 & 2\end{array}\right|=0 \quad$ [Applying $C_{2} \rightarrow C_{2}-7 C_{3}$ ]

$=\left|\begin{array}{ccc}a+2 x & b+2 y & c+2 z \\ a+2 x & b+2 y & c+2 z \\ x & y & z\end{array}\right| \quad$ [Applying $R_{1} \rightarrow R_{1}+2 R_{3}$ ]

$=\left|\begin{array}{ccc}0 & 0 & 0 \\ a+2 x & b+2 y & c+2 z \\ x & y & z\end{array}\right|=0 \quad$ [Applying $R_{1} \rightarrow R_{1}-R_{2}$ ]

(xii)

$\left|\begin{array}{lll}\left(2^{x}+2^{-x}\right)^{2} & \left(2^{x}-2^{-x}\right)^{2} & 1 \\ \left(3^{x}+3^{-x}\right)^{2} & \left(3^{x}-3^{-x}\right)^{2} & 1 \\ \left(4^{x}+4^{-x}\right)^{2} & \left(4^{x}-4^{-x}\right)^{2} & 1\end{array}\right|$

$=\left|\begin{array}{lll}\left(2^{2 x}+2^{-2 x}+2\right) & \left(2^{2 x}+2^{-2 x}-2\right) & 1 \\ \left(3^{2 x}+3^{-2 x}+2\right) & \left(3^{2 x}+3^{-2 x}-2\right) & 1 \\ \left(4^{2 x}+4^{-2 x}+2\right) & \left(4^{2 x}+4^{-2 x}-2\right) & 1\end{array}\right|$

$=\left|\begin{array}{lll}4 & \left(2^{2 x}+2^{-2 x}-2\right) & 1 \\ 4 & \left(3^{2 x}+3^{-2 x}-2\right) & 1 \\ 4 & \left(4^{2 x}+4^{-2 x}-2\right) & 1\end{array}\right| \quad\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}-C_{2}\right]$

$=4\left|\begin{array}{lll}1 & \left(2^{2 x}+2^{-2 x}-2\right) & 1 \\ 1 & \left(3^{2 x}+3^{-2 x}-2\right) & 1 \\ 1 & \left(4^{2 x}+4^{-2 x}-2\right) & 1\end{array}\right|$

$=0$

(Xiii)

$\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \cos (\alpha+\delta) \\ \sin \beta & \cos \beta & \cos (\beta+\delta) \\ \sin \gamma & \cos \gamma & \cos (\gamma+\delta)\end{array}\right|$

$=\left|\begin{array}{lll}\sin \alpha \sin \delta & \cos \alpha \cos \delta & \cos (\alpha+\delta) \\ \sin \beta \sin \delta & \cos \beta \cos \delta & \cos (\beta+\delta) \\ \sin \gamma \sin \delta & \cos \gamma \cos \delta & \cos (\gamma+\delta)\end{array}\right|$                            [Applying $C_{1} \rightarrow \sin \delta C_{1}$ and $C_{2} \rightarrow \cos \delta C_{2}$ ]

$=\left|\begin{array}{lll}\sin \alpha \sin \delta & \cos (\alpha+\delta) & \cos (\alpha+\delta) \\ \sin \beta \sin \delta & \cos (\beta+\delta) & \cos (\beta+\delta) \\ \sin \gamma \sin \delta & \cos (\gamma+\delta) & \cos (\gamma+\delta)\end{array}\right|$                                       [Applying $C_{2} \rightarrow C_{2}-C_{1}$ ]

$=0$

(xiv)

$\left|\begin{array}{ccc}\sin ^{2} 23^{\circ} & \sin ^{2} 67^{\circ} & \cos 180^{\circ} \\ -\sin ^{2} 67^{\circ} & -\sin ^{2} 23^{\circ} & \cos ^{2} 180^{\circ} \\ \cos 180^{\circ} & \sin ^{2} 23^{\circ} & \sin ^{2} 67^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 23^{\circ} & \sin ^{2}(90-23)^{\circ} & -1 \\ -\sin ^{2}(90-23)^{\circ} & -\sin ^{2} 23^{\circ} & 1 \\ -1 & \sin ^{2} 23^{\circ} & \sin ^{2}(90-23)^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} & -1 \\ -\cos ^{2} 23^{\circ} & -\sin ^{2} 23^{\circ} & 1 \\ -1 & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 23^{\circ}+\cos ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} & -1 \\ -\cos ^{2} 23^{\circ}-\sin ^{2} 23^{\circ} & -\sin ^{2} 23^{\circ} & 1 \\ -1+\sin ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right| \quad\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}+C_{2}\right]$

$=\left|\begin{array}{ccc}1 & 1 & -1 \\ -1 & -\sin ^{2} 23^{\circ} & 1 \\ -\cos ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right|$

$=(-1)\left|\begin{array}{ccc}-1 & 1 & -1 \\ 1 & -\sin ^{2} 23^{\circ} & 1 \\ \cos ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right|$

$=0$

(XVi)

$\left|\begin{array}{ccc}\sqrt{23}+\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{46} & 5 & \sqrt{10} \\ 3+\sqrt{115} & \sqrt{15} & 5\end{array}\right|$

$=\left|\begin{array}{ccc}\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5\end{array}\right|+\left|\begin{array}{ccc}\sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{46} & 5 & \sqrt{10} \\ \sqrt{115} & \sqrt{15} & 5\end{array}\right|$

$=\sqrt{3}\left|\begin{array}{ccc}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{5} & 5 & \sqrt{10} \\ \sqrt{3} & \sqrt{15} & 5\end{array}\right|+\sqrt{23}\left|\begin{array}{ccc}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{2} & 5 & \sqrt{10} \\ \sqrt{5} & \sqrt{15} & 5\end{array}\right|$

$=\sqrt{3} \times \sqrt{5}\left|\begin{array}{ccc}1 & 1 & \sqrt{5} \\ \sqrt{5} & \sqrt{5} & \sqrt{10} \\ \sqrt{3} & \sqrt{3} & 5\end{array}\right|+\sqrt{23} \times \sqrt{5}\left|\begin{array}{ccc}1 & \sqrt{5} & 5 & \sqrt{2} \\ \sqrt{5} & \sqrt{15} & \sqrt{5}\end{array}\right|$

$=0+0$

$=0$

(xvii)

$\left|\begin{array}{lll}\sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cot B & 1 \\ \sin ^{2} C & \cot C & 1\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} A-\sin ^{2} B & \cot A-\cot B & 0 \\ \sin ^{2} B & \cot B & 1 \\ \sin ^{2} C-\sin ^{2} B & \cot C-\cot B & 0\end{array}\right|$                    [Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{3} \rightarrow R_{3}-R_{2}$ ]

$=\left|\begin{array}{ccc}\sin ^{2} 23^{\circ} & \sin ^{2}(90-23)^{\circ} & -1 \\ -\sin ^{2}(90-23)^{\circ} & -\sin ^{2} 23^{\circ} & 1 \\ -1 & \sin ^{2} 23^{\circ} & \sin ^{2}(90-23)^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ} & -1 \\ -\cos ^{2} 23^{\circ} & -\sin ^{2} 23^{\circ} & 1 \\ -1 & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right|$                     $\left[\right.$ Applying $\left.C_{1} \rightarrow C_{1}+C_{2}\right]$

$=\left|\begin{array}{ccc}1 & 1 & -1 \\ -1 & -\sin ^{2} 23^{\circ} & 1 \\ -\cos ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right|$

$=(-1)\left|\begin{array}{ccc}-1 & 1 & -1 \\ 1 & -\sin ^{2} 23^{\circ} & 1 \\ \cos ^{2} 23^{\circ} & \sin ^{2} 23^{\circ} & \cos ^{2} 23^{\circ}\end{array}\right|$

$=0$

(XV)

$\left|\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ -\cos x & \sin x & -\cos y\end{array}\right|$

$=\frac{1}{\sin y \cos y}\left|\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & \cos 2 y \\ \sin x \sin y & \cos x \sin y & \sin ^{2} y \\ -\cos x \cos y & \sin x \cos y & -\cos ^{2} y\end{array}\right|$             [Applying $R_{2} \rightarrow \sin y R_{2}$ and $R_{3} \rightarrow \cos y R_{3}$ ]

$=\frac{1}{\sin y \cos y}\left|\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & \cos 2 y \\ \sin x \sin y-\cos x \cos y & \cos x \sin y+\sin x \cos y & \sin ^{2} y-\cos ^{2} y \\ -\cos x \cos y & \sin x \cos y & -\cos ^{2} y\end{array}\right|$ [Applying $R_{2} \rightarrow R_{2}+R_{3}$ ]

$=\frac{-1}{\sin y \cos y}\left|\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & \cos 2 y \\ \cos (x+y) & -\sin (x+y) & \cos 2 y \\ -\cos x \cos y & \sin x \cos y & -\cos ^{2} y\end{array}\right|$

$=0$

(Xvi)

$\left|\begin{array}{ccc}\sqrt{23}+\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{46} & 5 & \sqrt{10} \\ 3+\sqrt{115} & \sqrt{15} & 5\end{array}\right|$

$=\left|\begin{array}{ccc}\sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} & 5 & \sqrt{10} \\ 3 & \sqrt{15} & 5\end{array}\right|+\left|\begin{array}{ccc}\sqrt{23} & \sqrt{5} & \sqrt{5} \\ \sqrt{46} & 5 & \sqrt{10} \\ \sqrt{115} & \sqrt{15} & 5\end{array}\right|$

$=\sqrt{3}\left|\begin{array}{ccc}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{5} & 5 & \sqrt{10} \\ \sqrt{3} & \sqrt{15} & 5\end{array}\right|+\sqrt{23}\left|\begin{array}{ccc}1 & \sqrt{5} & \sqrt{5} \\ \sqrt{2} & 5 & \sqrt{10} \\ \sqrt{5} & \sqrt{15} & 5\end{array}\right|$

$=\sqrt{3} \times \sqrt{5}\left|\begin{array}{ccc}1 & 1 & \sqrt{5} \\ \sqrt{5} & \sqrt{5} & \sqrt{10} \\ \sqrt{3} & \sqrt{3} & 5\end{array}\right|+\sqrt{23} \times \sqrt{5}\left|\begin{array}{ccc}\sqrt{2} & 5 & \sqrt{2} \\ \sqrt{5} & \sqrt{15} & \sqrt{5}\end{array}\right|$

$=0+0$

$=0$

(xvii)

$=\left|\begin{array}{ccc}\sin ^{2} A-\sin ^{2} B & \cot A-\cot B & 0 \\ \sin ^{2} B & \cot B & 1 \\ \sin ^{2} C-\sin ^{2} B & \cot C-\cot B & 0\end{array}\right| \quad$ [Applying $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{3} \rightarrow R_{3}-R_{2}$ ]

$=\left|\begin{array}{ccc}\sin (A+B) \sin (A-B) & \frac{\cos A \sin B-\cos B \sin A}{\sin A \sin B} & 0 \\ \sin ^{2} B & \cot B & 1 \\ \sin (C+B) \sin (C-B) & \frac{\cos C \sin B-\cos B \sin C}{\sin B \sin C} & 0\end{array}\right|$

$=\left|\begin{array}{ccc}\sin (\pi-C) \sin (A-B) & \frac{-\sin (A-B)}{\sin A \sin B} & 0 \\ \sin ^{2} B & \cot B & 1 \\ \sin (\pi-A) \sin (C-B) & \frac{-\sin (C-B)}{\sin B \sin C} & 0\end{array}\right| \quad[\because A+B+C=\pi]$

$=\left|\begin{array}{ccc}\sin C \sin (A-B) & \frac{-\sin (A-B)}{\sin A \sin B} & 0 \\ \sin ^{2} B & \frac{\cos B}{\sin B} & 1 \\ \sin A \sin (C-B) & \frac{-\sin (C-B)}{\sin B \sin C} & 0\end{array}\right|$

$=\frac{\sin (A-B) \sin (C-B)}{\sin B}\left|\begin{array}{ccc}\sin C & \frac{-1}{\sin A} & 0 \\ \sin ^{2} B & \cos B & 1 \\ \sin A & \frac{-1}{\sin C} & 0\end{array}\right|$

$=\frac{\sin (A-B) \sin (C-B)}{\sin B \sin A \sin C}\left|\begin{array}{ccc}\sin C \sin A & -1 & 0 \\ \sin ^{2} B & \cos B & 1 \\ \sin A \sin C & -1 & 0\end{array}\right| \quad$ [Applying $R_{1} \rightarrow \sin A R_{1}$ and $R_{3} \rightarrow \sin C R_{3}$ ]

$=\frac{\sin (A-B) \sin (C-B)}{\sin B \sin A \sin C}\left|\begin{array}{ccc}0 & 0 & 0 \\ \sin ^{2} B & \cos B & 1 \\ \sin A \sin C & -1 & 0\end{array}\right| \quad$ [Applying $R_{1} \rightarrow R_{1}-R_{3}$ ]

$=0$