Section 1
Properties Of Determinants
Evaluate the following determinants:(i) $\left|\begin{array}{cc}-2 & 3 \\ 4 & -9\end{array}\right|$(ii) $\begin{array}{ll}\frac{1}{2} & 8 \\ 4 & 2\end{array} \mid$(iii) $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$(iv) $\left|\begin{array}{ll}\sec \theta & \tan \theta \\ \tan \theta & \sec \theta\end{array}\right|$(v) $\left|\begin{array}{cc}2+\sqrt{3} & -5 \\ 2 & 2-\sqrt{3}\end{array}\right|$(vi) $\left|\begin{array}{cc}2+3 i & 4 \\ 1 & 2-3 i\end{array}\right|$(vii) $\begin{array}{ll}\sqrt{5} & \sqrt{12} \\ \sqrt{3} & \sqrt{20}\end{array} \mid$(viii) $\begin{array}{cc}\log _{a} b & 1 \\ 1 & \log _{b} a\end{array} \mid$(ix) $\left|\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|$
Find the value of $x$ if(i) $\left|\begin{array}{cc}x & x \\ 4 & 2 x\end{array}\right|=0$(ii) $\left|\begin{array}{cc}4 x+2 & 2 x+1 \\ x+1 & 2 x-1\end{array}\right|=0$(iii) $\left|\begin{array}{cc}2 x-4 & x+2 \\ x-2 & 2 x+4\end{array}\right|=0$(iv) $\left|\begin{array}{cc}x-1 & x-2 \\ x & x-3\end{array}\right|=0$(v) $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$(vi) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$
Show that $\left|\begin{array}{ll}a d+b c & b d-a c \\ a c-b d & a d+b c\end{array}\right|=\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)$
Evaluate the following determinants:(i) $\left|\begin{array}{ccc}2 & 4 & 7 \\ 3 & 6 & 9 \\ 4 & 8 & 11\end{array}\right|$(ii) $\left|\begin{array}{rrr}5 & 1 & 0 \\ 2 & 3 & -1 \\ -3 & 2 & 0\end{array}\right|$(iii) $\left|\begin{array}{ccc}1 & x & y \\ 0 & \cos x & \sin y \\ 0 & \sin x & \cos y\end{array}\right|$(iv) $\left|\begin{array}{ccc}\sin \theta & 1 & 0 \\ 0 & \cos \phi & -\cos \theta \\ \sin \phi & 0 & 1\end{array}\right|$.
(i) If $A=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]$, then show that $|2 A|=4|A|$.(ii) If $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{array}\right]$, then show that $|3 A|=27|A|$.
Find the value of $x$ if(i) $\left|\begin{array}{lll}x & 3 & 3 \\ 3 & 3 & x \\ 2 & 3 & 3\end{array}\right|=0$(ii) $\left|\begin{array}{lll}x^{2} & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{array}\right|=28$
Write the minor and cofactor of each element of the following determinants and also evaluate the determinant in each case:(i) $\left|\begin{array}{rr}5 & -10 \\ 0 & 3\end{array}\right|$(ii) $\left|\begin{array}{rrr}1 & 3 & -2 \\ 4 & -5 & 6 \\ 3 & 5 & 2\end{array}\right|$(iii) $\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$(iv) $\left|\begin{array}{rrr}1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2\end{array}\right|$
Verify that $|A B|=|A||B|$ if $A=\left[\begin{array}{rr}3 & 0 \\ -1 & 4\end{array}\right]$ and $B=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right]$.
Verify that $|A B|=|A||B|$ if $A=\left[\begin{array}{rrr}3 & -2 & 1 \\ 1 & 0 & 4 \\ 1 & -1 & 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-1 & 1 & 2 \\ 3 & 1 & -2 \\ 1 & 0 & 1\end{array}\right]$.
Show that $A=\left[\begin{array}{rrr}1 & -2 & 3 \\ 1 & 2 & 1 \\ -1 & 2 & -3\end{array}\right]$ is a singular matrix.
Find the value of $x$ if the matrix $A=\left[\begin{array}{rrr}4 & 3 & 5 \\ 3 & -2 & 7 \\ 10 & -1 & x\end{array}\right]$ is singular.
If $\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\ 18 & 6\end{array}\right|$, then $x$ is equal to(a) 6(b) $\pm 6$(c) $-6$(d) 6,6
Using cofactors of elements of second row, evaluate $\Delta=\left|\begin{array}{ccc}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{array}\right|$.
Using cofactors of elements of third column, evaluate $\Delta=\left|\begin{array}{ccc}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|$.
If $\Delta=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$ and $A_{i j}$ is the cofactor of $a_{i j}$, then value of $\Delta$ is given by(a) $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}$(b) $a_{11} A_{11}+a_{12} A_{21}+a_{13} A_{31}$(c) $a_{21} A_{11}+a_{22} A_{12}+a_{23} A_{13}$(d) $a_{11} A_{11}+a_{21} A_{21}+a_{31} A_{31}$
If $A=2 B$, where $A$ and $B$ are square matrices of order $3 \times 3$ and $|B|=5$, what is $|A| ?$
If $A, B, C$ are angles of a triangle. Find the value of $\Delta$ if$$\Delta=\left|\begin{array}{ccc}\sin (A+B+C) & \sin B & \cos C \\-\sin B & 0 & \tan A \\\cos (A+B) & -\tan A & 0\end{array}\right|$$
(i) If $A=\left[\begin{array}{rr}4 & -2 \\ 2 & 1\end{array}\right]$, find $|5 A|$.(ii) If $A=\left[\begin{array}{rr}3 & -4 \\ 1 & 2\end{array}\right]$, find $2|A|$.