Solution: We have $A=\left(\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right), B=\left(\begin{array}{cc}1 & -3 \\ -2 & 4\end{array}\right)$ and $C=\left(\begin{array}{cc}-4 & 6 \\...
If and , find .
Solution: We have $A=\left(\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right), B=\left(\begin{array}{cc}0 & 4 \\ -1 & 7\end{array}\right)$. (i) Let's compute first $A^{2}$ $A^{2}=A...
Give an example of three matrices A, B, C such that AB = AC but B ≠ C.
Solution: We have $\boldsymbol{A B}=\boldsymbol{A} \boldsymbol{C}$ but $\boldsymbol{B} \neq \boldsymbol{C}$. WE need to find: $\boldsymbol{A}, \boldsymbol{B}$. Let's take...
Given an example of two matrices A and B such that A ≠ O, B ≠ O, AB = O and BA ≠ O.
Solution: We have $\boldsymbol{A} \neq \boldsymbol{O}, \boldsymbol{B} \neq \boldsymbol{O}, \boldsymbol{A B}=\boldsymbol{O}$ and $\boldsymbol{B A} \neq \boldsymbol{O}$. We need to find:...
If , prove that for all .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{1} & \mathbf{1} \\ \mathbf{0} & \mathbf{1}\end{array}\right)$. We need to show: $A^{n}=\left(\begin{array}{ll}1 & n \\ 0...
If , find , where .
Solution: We have $A=\left(\begin{array}{cc}3 & 4 \\ -4 & -3\end{array}\right)$ and equation $f(x)=x^{2}-5 x+7$. (i) Let us compute first $A^{2}$ $A^{2}=A A=\left(\begin{array}{cc} 3 & 4...
Find the values of and for which
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\boldsymbol{a} & \boldsymbol{b} \\ -\boldsymbol{a} & \boldsymbol{2} \boldsymbol{b}\end{array}\right),...
If , find
Solution: We have $A=\left(\begin{array}{ccc}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right), B=\left(\begin{array}{lll}x & 4 & 1\end{array}\right)$ and...
If , find
Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 2 & 5\end{array}\right), B=\left(\begin{array}{lll}1 & x & 1\end{array}\right)$ and...
If , show that
Solution: We have $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$ and to show $A^{2}=$ $\left(\begin{array}{cc}\cos ^2 \alpha & \sin...
If , show that
Solution: We have $\boldsymbol{F}(\boldsymbol{X})=\left(\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right)$ and to show...
If and then find the values of and .
Solution: We have $A=\left(\begin{array}{ll}1 & -1 \\ 2 & -1\end{array}\right), B=\left(\begin{array}{ll}a & -1 \\ b & -1\end{array}\right)$ and $(A+B)^{2}=\left(A^{2}+\right.$...
Find the matrix A such that A. .
Solution: We have $\boldsymbol{B}=\left(\begin{array}{ll}\mathbf{2} & \mathbf{3} \\ \mathbf{4} & \mathbf{5}\end{array}\right)$ and $\boldsymbol{C}=\left(\begin{array}{cc}\mathbf{0} &...
Find the matrix A such that .
Solution: We have $\boldsymbol{B}=\left(\begin{array}{cc}\mathbf{5} & -\mathbf{7} \\ -\mathbf{2} & \mathbf{3}\end{array}\right)$ and $\boldsymbol{C}=\left(\begin{array}{cc}-\mathbf{1 6}...
If , find the value of a and such that
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & \mathbf{2} \\ \mathbf{1} & \mathbf{1}\end{array}\right)$. To find $\boldsymbol{a}, \boldsymbol{b}$ such that...
If , find and such that .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & \mathbf{1} \\ \mathbf{7} & \mathbf{5}\end{array}\right)$. To find $\boldsymbol{x}, \boldsymbol{y}$ such that...
Solve for and , when
Solution: We have $A=\left(\begin{array}{cc}3 & -4 \\ 1 & 2\end{array}\right), B=\left(\begin{array}{c}3 \\ 11\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y\end{array}\right)$. We...
Find the values of and , when
Solution: We have $A=\left(\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right), B=\left(\begin{array}{l}1 \\ 3\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y\end{array}\right)$. To...
If and , find .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{1} & \mathbf{2} \\ \mathbf{4} & \mathbf{- 3}\end{array}\right)$. Now addition/subtraction of two matrices is possible if...
If , find , where
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)$. Now addition/subtraction of two matrices is possible if order of both the matrices are same and...
If , find so that .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{3} & -\mathbf{2} \\ \mathbf{4} & -\mathbf{2}\end{array}\right)$. Now addition/subtraction of two matrices is possible if...
Show that the matrix satisfies the equation .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{2} & \mathbf{3} \\ \mathbf{1} & \mathbf{2}\end{array}\right)$. Now addition of two matrices is possible if order of both the...
If , show that .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{3} & \mathbf{1} \\ -\mathbf{1} & \mathbf{2}\end{array}\right)$. Now addition of two matrices is possible if order of both the...
If then find
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 2} \\ -\mathbf{3} & \mathbf{4}\end{array}\right)$. Now addition of two matrices is possible if order of both...
If and , find .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{2}\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{cc}\mathbf{0} &...
If , show that
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{4} & -\mathbf{1} & -\mathbf{4} \\ \mathbf{3} & \mathbf{0} & -\mathbf{4} \\ \mathbf{3} & -\mathbf{1} &...
If , show that
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & \mathbf{- 2} & -\mathbf{4} \\ \mathbf{- 1} & \mathbf{3} & \mathbf{4} \\ \mathbf{1} & -\mathbf{2} &...
If , show that .
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\boldsymbol{a} \boldsymbol{b} & \boldsymbol{b}^{2} \\ -\boldsymbol{a}^{2} & -\boldsymbol{a} \boldsymbol{b}\end{array}\right) .$ To...
If and verify that
Solution: We have $A=\left(\begin{array}{ccc}1 & 0 & 2 \\ 3 & -1 & 0 \\ -2 & 1 & 1\end{array}\right), B=\left(\begin{array}{ccc}0 & 5 & -4 \\ -2 & 1 & 3 \\ 1...
Verify that , when
Solution: We have $A=\left(\begin{array}{cc}2 & 3 \\ -1 & 4 \\ 0 & 1\end{array}\right), B=\left(\begin{array}{cc}5 & -3 \\ 2 & 1\end{array}\right)$. and...
Verify that , when
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{1} & \mathbf{2} \\ \mathbf{3} & \mathbf{4}\end{array}\right), \boldsymbol{B}=\left(\begin{array}{cc}\mathbf{2} &...
For the following matrices, verify that :
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & \mathbf{3} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} & \mathbf{2}\end{array}\right),...
For the following matrices, verify that :
Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 5 \\ 0 & 1 & 3\end{array}\right), B=\left(\begin{array}{ccc}2 & 3 & 0 \\ 1 & 0 & 4 \\ 1 & -1 &...
If and . show that is a zero matrix.
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \boldsymbol{c} & -\boldsymbol{b} \\ -\boldsymbol{c} & \mathbf{0} & \boldsymbol{a} \\ \boldsymbol{b} &...
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{2} & -\mathbf{3} & -\mathbf{5} \\ -\mathbf{1} & \mathbf{4} & \mathbf{5} \\ \mathbf{1} & -\mathbf{3} &...
Show that in each of the following cases:
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{1} & \mathbf{3} & -\mathbf{1} \\ \mathbf{2} & \mathbf{2} & -\mathbf{1} \\ \mathbf{3} & \mathbf{0} &...
Show that in each of the following cases:
Solution: We have $A=\left(\begin{array}{lll}1 & 2 & 1 \\ 3 & 4 & 2 \\ 1 & 3 & 2\end{array}\right)$ and $B=\left(\begin{array}{ccc}10 & -4 & -1 \\ -11 & 5 & 0...
Show that in each of the following cases:
Solution: We have $A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$ and $B=\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi...
Show that in each of the following cases:
Solution: We have $\boldsymbol{A}=\left(\begin{array}{lll}\mathbf{1} & \mathbf{2} & \mathbf{3} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{1} & \mathbf{1} &...
Show that in each of the following cases :
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{5} & -\mathbf{1} \\ \mathbf{6} & \mathbf{7}\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{ll}\mathbf{2} &...
Compute and , which ever exists when
Solution: We have $\boldsymbol{B}=\left(\begin{array}{ccc}\mathbf{1} & \mathbf{0} & \mathbf{1} \\ -\mathbf{1} & \mathbf{2} & \mathbf{1}\end{array}\right)$ and...
Compute and BA, which ever exists when
Solution: We have $\boldsymbol{A}=\left(\begin{array}{llll}\mathbf{1} & 2 & 3 & 4\end{array}\right)$ and $\boldsymbol{B}=\left(\begin{array}{l}\mathbf{1} \\ \mathbf{2} \\ \mathbf{3} \\...
Compute AB and BA, which ever exists when
Solution: We have $\boldsymbol{A}=\left(\begin{array}{ccc}\mathbf{0} & \mathbf{1} & -\mathbf{5} \\ \mathbf{2} & \mathbf{4} & \mathbf{0}\end{array}\right)$ and...
Compute AB and BA, which ever exists when
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}-\mathbf{1} & \mathbf{1} \\ -\mathbf{2} & \mathbf{2} \\ -\mathbf{3} & \mathbf{3}\end{array}\right)$ and...
Compute AB and BA, which ever exists when
Solution: We have $\boldsymbol{A}=\left(\begin{array}{cc}\mathbf{2} & \mathbf{- 1} \\ \mathbf{3} & \mathbf{0} \\ \mathbf{- 1} & \mathbf{4}\end{array}\right)$ and...