Solution: Given that $A=\left[\begin{array}{cc} 3 & 1 \\ -1 & 2 \end{array}\right]$ $A^{2}=\left[\begin{array}{cc} 3 & 1 \\ -1 & 2 \end{array}\right]\left[\begin{array}{cc} 3 & 1...
Show that satisfies the equation Hence find
Solution: Given that $A=\left[\begin{array}{cc}-8 & 5 \\ 2 & 4\end{array}\right]$ $A^{2}=\left[\begin{array}{cc}-8 & 5 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}-8 & 5 \\...
If verify that , where and . Hence find .
Solution: On considering, $A^{2}=\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right]\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right]=\left[\begin{array}{ll} 4+3 &...
Let and
. Show that
(i)
Solution: (i) To show that $[F(\alpha) G(\beta)]^{-1}=G(-\beta) F(-\alpha)$ It is already known that $[G(\beta)]^{-1}=G(-\beta)$ $[F(\alpha)]^{-1}=F(-\alpha)$ $\text { And } L H S=[F(\alpha)...
Let and
. Show that
(i)
(ii)
Solution: (i) Given that $\mathrm{F}(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$...
Given .Compute
Solution: Given that $A=\left[\begin{array}{lll}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{array}\right]$ and $B^{-1}=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4...
Find the inverse of the matrix , and show that .
Solution: $A=\left[\begin{array}{cc}a & b \\ c & \frac{1+b c}{a}\end{array}\right]$ Now, $|\mathrm{A}|=\frac{\mathrm{a}+\mathrm{abc}}{\mathrm{a}}-{...
If , then show that .
Solution: $\begin{aligned} &A=\left[\begin{array}{ll} 4 & 5 \\ 2 & 1 \end{array}\right] \\ &|A|=4-10=-6 \neq 0 \\ &\operatorname{adj} A=\left[\begin{array}{cc} 1 & -5 \\ -2...
Given , compute and show that
Solution: Given that $A=\left[\begin{array}{cc} 2 & -3 \\ -4 & 7 \end{array}\right]$ $|\mathrm{A}|=14-12=2 \neq 0$ $\operatorname{adj} A=\left[\begin{array}{ll} 7 & 3 \\ 4 & 2...
Let and . Find
Solution: It is given that $A=\left[\begin{array}{ll} 3 & 2 \\ 7 & 5 \end{array}\right]$ $|A|=15-14=1 \neq 0$ So adj $A=\left[\begin{array}{cc}5 & -2 \\ -7 & 3\end{array}\right]$...
For the following pair of matrices verify that .
(i) and
(ii) and
Solution (i) Given that $A=\left[\begin{array}{ll}3 & 2 \\ 7 & 5\end{array}\right]$ $|A|=1 \neq 0$ Therefore, adj $A=\left[\begin{array}{cc}5 & -2 \\ -7 & 3\end{array}\right]$...
Find the inverse of each of the following matrices and verify that .
(i)
(ii)
Solution: (i) We have $|\mathrm{A}|=1\left|\begin{array}{ll}4 & 3 \\ 3 & 4\end{array}\right|-3\left|\begin{array}{ll}1 & 3 \\ 1 & 4\end{array}\right|+3\left|\begin{array}{ll}1 &...
Find the inverse of each of the following matrices.
Solution: (i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero. $|\mathrm{A}|=1\left|\begin{array}{cc}\cos \alpha & \sin \alpha \\...
Find the inverse of each of the following matrices.
Solution: (i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero. $|\mathrm{A}|=0\left|\begin{array}{ll}-3 & 0 \\ -3 &...
Find the inverse of each of the following matrices.
Solution: (i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero. $|\mathrm{A}|=2\left|\begin{array}{cc}2 & -1 \\ -1 &...
Find the inverse of each of the following matrices.
Solution: (i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero. $|\mathrm{A}|=1\left|\begin{array}{ll}3 & 1 \\ 1 &...
Find the inverse of each of the following matrices:
(i)
(ii)
Solution: (i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero. $\left.\begin{array}{l} \text { Now,...
Find the inverse of each of the following matrices:
(i)
(ii)
Solution: (i) The criteria of existence of inverse matrix is the determinant of a given matrix should not be equal to zero. Now, $|A|=\cos \theta(\cos \theta)+\sin \theta(\sin \theta)$ $=1$...
Find
Solution: Given that $\begin{aligned} &A=\left[\begin{array}{ccc} 1 & -2 & 3 \\ 0 & 2 & -1 \\ -4 & 5 & 2 \end{array}\right] \\ &\text { Cofactors of } A \text { are }...
, show that
Solution: Given that $A=\left[\begin{array}{ccc}-1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{array}\right]$ Cofactors of $A$ are $C_{11}=-3$ $C_{21}=6$ $C_{31}=6$ $C_{12}=-6$...
If , show that
Solution: Given that $A=\left[\begin{array}{ccc}-4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{array}\right]$ Cofactors of $A$ $C_{11}=-4$ $C_{21}=-3$ $C_{31}=-3$...
For the matrix , show that
Solution: It is given that $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 3 & 0 \\ 18 & 2 & 10\end{array}\right]$ Therefore cofactors of $A$ $C_{11}=30$ $C_{21}=12$ $C_{31}=-3$...
Compute the adjoint of each of the following matrices.
(i)
(ii)
Solution: (i) Suppose $A=\left[\begin{array}{ccc}2 & -1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & -1\end{array}\right]$ Therefore cofactors of $A$ $\mathrm{C}_{11}=-22$ $C_{21}=11$...
Compute the adjoint of each of the following matrices.
(i)
(ii)
Solution: (i) Suppose $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$ Therefore cofactors of $A$ are $C_{11}=-3$ $C_{21}=2$ $C_{31}=2$...
Find the adjoint of each of the following matrices:
(i)
(ii) Verify that for the above matrices.
Solution: (i) Suppose $A=\left[\begin{array}{ll}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$ Therefore cofactors of $A$ are $C_{11}=\cos \alpha$ $C_{12}=-\sin...
Find the adjoint of each of the following matrices:
(i)
(ii) Verify that for the above matrices.
Solution: (i) Suppose $A=\left[\begin{array}{cc}-3 & 5 \\ 2 & 4\end{array}\right]$ Cofactors of $A$ are $C_{11}=4$ $C_{12}=-2$ $C_{21}=-5$ $C_{22}=-3$ Since, adj...