Chapter 10

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{cccc} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{array}\right|=a b c d\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{cccc} 1+x & 2 & 3 & 4 \\ 1 & 2+x & 3 & 4 \\ 1 & 2 & 3+x & 4 \\ 1 & 2 & 3 & 4+x \end{array}\right|=x^{3}(x+10) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{llll} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{array}\right|=(x+3 a)(x-a)^{3} . $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a x & b y & c z \\ x^{2} & y^{2} & z^{2} \\ 1 & 1 & 1 \end{array}\right|=\left|\begin{array}{ccc} a & b & c \\ x & y & z \\ y z & z x & x y \end{array}\right| . $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{cccc} x^{3} & 3 x^{2} & 3 x & 1 \\ x^{2} & x^{2}+2 x & 2 x+1 & 1 \\ x & 2 x+1 & x+2 & 1 \\ 1 & 3 & 3 & 1 \end{array}\right|=(x-1)^{6} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} (x-2)^{2} & (x-1)^{2} & x^{2} \\ (x-1)^{2} & x^{2} & (x+1)^{2} \\ x^{2} & (x+1)^{2} & (x+2)^{2} \end{array}\right|=-8 $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} (x-2)^{2} & (x-1)^{2} & x^{2} \\ (x-1)^{2} & x^{2} & (x+1)^{2} \\ x^{2} & (x+1)^{2} & (x+2)^{2} \end{array}\right|=-8 $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{cccc} 0 & x & y & z \\ -x & 0 & c & b \\ -y & -c & 0 & a \\ -z & -b & -a & 0 \end{array}\right|=(a x-b y+c z)^{2} . $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a & b-c & c+b \\ a+c & b & c-a \\ a-b & b+a & c \end{array}\right|=(a+b+c)\left(a^{2}+b^{2}+c^{2}\right) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} (b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2} \end{array}\right|=\left|\begin{array}{ccc} (a+b)^{2} & c a & c b \\ c a & (b+c)^{2} & a b \\ b c & a b & (c+a)^{2} \end{array}\right|=2 a b c(a+b+c)^{3} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} 1+a^{2}-b^{2} & 2 a b & -2 b \\ 2 a b & 1-a^{2}+b^{2} & 2 a \\ 2 b & -2 a & 1-a^{2}-b^{2} \end{array}\right|=\left(1+a^{2}+b^{2}\right)^{3} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2} \end{array}\right|=4\left|\begin{array}{ccc} a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1 \end{array}\right| $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} -b c & b^{2}+b c & c^{2}+b c \\ a^{2}+a c & -a c & c^{2}+a c \\ a^{2}+a b & b^{2}+a b & -a b \end{array}\right|=(b c+c a+a b)^{3} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a x-b y-c z & a y+b x & c x+a z \\ a y+b x & b y-c z-a x & b z+c y \\ c x+a z & b z+c y & c z-a x-b y \end{array}\right|=\left(x^{2}+y^{2}+z^{2}\right)\left(a^{2}+b^{2}+c^{2}\right)(a x+b y+c z) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} 1 & b c+a d & b^{2} c^{2}+a^{2} d^{2} \\ 1 & c a+b d & c^{2} a^{2}+b^{2} d^{2} \\ 1 & a b+c d & a^{2} b^{2}+c^{2} d^{2} \end{array}\right|=(a-b)(a-c)(a-d)(b-c)(b-d)(c-d) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} x^{2} & x^{2}-(y-z)^{2} & y z \\ y^{2} & y^{2}-(z-x)^{2} & z x \\ z^{2} & z^{2}-(x-y)^{2} & x y \end{array}\right|=(x-y)(y-z)(z-x)(x+y+z)\left(x^{2}+y^{2}+z^{2}\right) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} a_{1} \alpha_{1}+b_{1} \beta_{1} & a_{1} \alpha_{2}+b_{1} \beta_{2} & a_{1} \alpha_{3}+b_{1} \beta_{3} \\ a_{2} \alpha_{1}+b_{2} \beta_{1} & a_{2} \alpha_{2}+b_{2} \beta_{2} & a_{2} \alpha_{3}+b_{2} \beta_{3} \\ a_{3} \alpha_{1}+b_{3} \beta_{1} & a_{3} \alpha_{2}+b_{3} \beta_{2} & a_{3} \alpha_{3}+b_{3} \beta_{3} \end{array}\right|=0 $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} \cos (A-P) & \cos (A-Q) & \cos (A-R) \\ \cos (B-P) & \cos (B-Q) & \cos (B-R) \\ \cos (C-P) & \cos (C-Q) & \cos (C-R) \end{array}\right|=0 $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} y z-x^{2} & z x-y^{2} & x y-z^{2} \\ z x-y^{2} & x y-z^{2} & y z-x^{2} \\ x y-z^{2} & y z-x^{2} & z x-y^{2} \end{array}\right|=\left|\begin{array}{ccc} r^{2} & u^{2} & u^{2} \\ u^{2} & r^{2} & u^{2} \\ u^{2} & u^{2} & r^{2} \end{array}\right|, \text { where } r^{2}=x^{2}+y^{2}+z^{2} \text { and } u^{2}=y z+z x+x y $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} \cos ^{2} x & \cos x \sin x & -\sin x \\ \cos x \sin x & \sin ^{2} x & \cos x \\ \sin x & -\cos x & 0 \end{array}\right|=1 $$

If \(a^{-1}+b^{-1}+c^{-1}=0\), prove that \(\left|\begin{array}{ccc}1+a & 1 & 1 \\\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right|=a b c\).

If \(p+q+r=0\), prove that \(\left|\begin{array}{lll}p a & q b & r c \\ q c & r a & p b \\ r b & p c & q a\end{array}\right|=p q r\left|\begin{array}{lll}a & b & c \\ c & a & b \\ b & c & a\end{array}\right|\).

If \(2 s=a+b+c\), prove that \(\left|\begin{array}{ccc}a^{2} & (s-a)^{2} & (s-a)^{2} \\ (s-b)^{2} & b^{2} & (s-b)^{2} \\ (s-c)^{2} & (s-c)^{2} & c^{2}\end{array}\right|=2 s^{3}(s-a)(s-b)(s-c) .\)

Without expanding at any stage show that \(\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=x A+B\), where \(A\) and \(B\) are determinants of order 3 not involving \(x\).

If \(y=\sin p x\) and \(y_{r}\) means \(r\) th derivative of \(y\) then prove that \(\left|\begin{array}{lll}y & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8}\end{array}\right|=0\).

If \(a, b, c\) (all +ive) are the \(p\) th, \(q\) th, \(r\) th, terms respectively of a geometric progression, then prove that \(\left|\begin{array}{lll}\log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1\end{array}\right|=0\)

If \(a, b, c\) are given to be in A.P., prove that \(\left|\begin{array}{ccc}x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c\end{array}\right|=0\).

Given that \(A+B+C=\pi\), prove that \(\left|\begin{array}{lll}\sin ^{2} A & \sin A \cos A & \cos ^{2} A \\ \sin ^{2} B & \sin B \cos B & \cos ^{2} B \\ \sin ^{2} C & \sin C \cos C & \cos ^{2} C\end{array}\right|=-\sin (A-B) \sin (B-C) \sin (C-A)\).

Find the roots of the equation \(\left|\begin{array}{ccc}x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1\end{array}\right|=0 .\) \\{Ans. \(\left.-1.2\right\\}\)

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{lll} a & a & x \\ x & x & x \\ b & x & b \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{lll} x+a & a^{2} & a^{3} \\ x+b & b^{2} & b^{3} \\ x+c & c^{2} & c^{3} \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} 15-2 x & 11 & 10 \\ 11-3 x & 17 & 16 \\ 7-x & 14 & 13 \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} x-2 & 2 x-3 & 3 x-4 \\ x-4 & 2 x-9 & 3 x-16 \\ x-8 & 2 x-27 & 3 x-64 \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} 4 x & 6 x+2 & 8 x+1 \\ 6 x+2 & 9 x+3 & 12 x \\ 8 x+1 & 12 x & 16 x+2 \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} 3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8 \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{ccc} x+2 & 2 x+3 & 3 x+4 \\ 2 x+3 & 3 x+4 & 4 x+5 \\ 3 x+5 & 5 x+8 & 10 x+17 \end{array}\right|=0 $$

EQUATIONS CONTAINING DETERMINANTS. $$ \left|\begin{array}{lll} x+2 & x+6 & x-1 \\ x+6 & x-1 & x+2 \\ x-1 & x+2 & x+6 \end{array}\right|=0 $$

Show that \(x=2\) is a root of \(\left|\begin{array}{ccc}x & -6 & -1 \\ 2 & -3 x & x-3 \\ -3 & 2 x & x+2\end{array}\right|=0\) and solve it completely.

Solve \(\left|\begin{array}{ccc}1 & 3 & 9 \\ 1 & x & x^{2} \\ 4 & 6 & 9\end{array}\right|=0\)

Show that \(x=-9\) is a root of \(\left|\begin{array}{lll}x & 3 & 7 \\ 2 & x & 2 \\\ 7 & 6 & x\end{array}\right|=0\) and find the other two roots.

Given \(a+b+c=0\), solve \(\left|\begin{array}{ccc}a-x & c & b \\ c & b-x & a \\\ b & a & c-x\end{array}\right|=0\)

If \((b-c)^{2} \neq(a-b)(c-a)\), solve for \(x\left|\begin{array}{lll}a+x & b+x & c+x \\ b+x & c+x & a+x \\ c+x & a+x & b+x\end{array}\right|=0\)

If \(a \neq b \neq c\), solve for \(x\left|\begin{array}{ccc}0 & x-a & x-b \\\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|=0\).

If \(a \neq p, b \neq q, c \neq r\) and \(\left|\begin{array}{lll}p & b & c \\ a & q & c \\ a & b & r\end{array}\right|=0\), then find the value of \(\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c} .\)

Suppose three digit numbers \(A 28,3 B 9\) and \(62 C\) where \(A, B\) and \(C\) are integers between 0 and 9 , are divisible by a fixed integer \(k\). Prove that the determinant \(\left|\begin{array}{lll}A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{array}\right|\) is also divisible by \(k\).

For a fixed positive integer \(n\), if \(D=\mid \begin{array}{ccc}n ! & (n+1) ! & (n+2) ! \\ (n+1) ! & (n+2) ! & (n+3) ! \\ (n+2) ! & (n+3) ! & (n+4) !\end{array}\) then show that \(\left[\frac{D}{(n !)^{3}}-4\right]\) is divisible by \(n\).

If \(x, y\) and \(z\) are all different and given that \(\left|\begin{array}{lll}x & x^{2} & 1+x^{3} \\ y & y^{2} & 1+y^{3} \\ z & z^{2} & 1+z^{3}\end{array}\right|=0\), prove that \(1+x y z=0\)

If \(x, y\) and \(z\) are all different and given that \(\left|\begin{array}{lll}x & x^{3} & x^{4}-1 \\ y & y^{3} & y^{4}-1 \\ z & z^{3} & z^{4}-1\end{array}\right|=0\), prove that \(x y z(x y+y z+z x)=x+y+z\).

Let \(\alpha\) be a repeated root of quadratic equation \(f(x)=0\) and \(A(x), B(x), C(x)\) be polynomials of degree 3,4 and 5 respectively, then show that \(\left|\begin{array}{lll}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A^{\prime}(\alpha) & B^{\prime}(\alpha) & C^{\prime}(\alpha)\end{array}\right|\) is divisible by \(f(x)\), where ' denotes the derivative.