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Verify that $A^2=I\;when\;A=\begin{bmatrix}0 & 1 & 1\\4 & 3 & 4\\3 & 3 & 4\end{bmatrix}.$

asked Dec 23, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}0 & x\\x & 0\end{bmatrix},B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\;and\;x^2=-1,then\;show\;that\;(A+B)^2=A^2+B^2.$

asked Dec 23, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{bmatrix},\;then\;show\;that\;A^2=\begin{bmatrix}\cos2\theta & \sin2\theta\\-\sin2\theta & \cos2\theta\end{bmatrix}.$

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that$:\[(i)\;(A-B)^T=A^T-B^T.\]

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that$:\[(h)\;(A-B)C=AC-BC.\]

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;$and$\;a=4,b=-2$.Show that$\;(g)\;(AB)^T=B^TA^T.$

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that$:\[(f)\;(bA)^T=bA^T.\]

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that$:\[(e)\;(A^T)^T=A.\]

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that$:\[(d)\;a(C-A)=aC-aA.\]

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;,\;a=4,b=-2\;$.Show that:$\;(c)\;(a+b)B=aB+bB$

asked Dec 23, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that:$\[(b)\;A(BC)=(AB)C\]

asked Dec 22, 2012 by sreemathi.v

1 answer

Let $A=\begin{bmatrix}1 & 2\\1 & 3\end{bmatrix},B=\begin{bmatrix}4 & 0\\1 & 5\end{bmatrix},C=\begin{bmatrix}2 & 0\\1 & 2\end{bmatrix}\;and\;a=4,b=-2.Show\;that:$\[(a)\;A+(B+C)=(A+B)+C\]

asked Dec 22, 2012 by sreemathi.v

1 answer

Show that if A and B are square matrices such that AB=BA,then\[(A+B)^2=A^2+2AB+B^2\]

asked Dec 22, 2012 by sreemathi.v

1 answer

Let A and B be the square matrices of the order $3\times 3$.Is $(AB)^2=A^2B^2?$ Give reasons

asked Dec 22, 2012 by sreemathi.v

1 answer

Show that $A'A$ and $AA'$ are both symmetric matrices for any matrix A

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}1 & 2\\4 & 1\\5 & 0\end{bmatrix},B=\begin{bmatrix}1 & 2\\6 & 4\\7 & 3\end{bmatrix},then\;verify\;that:(i)\quad(2A+B)'=2A'+B'$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}0 & -1 & 2\\4 & 3 & -4\end{bmatrix}\;and\;B=\begin{bmatrix}4 & 0\\1 & 3\\2 & 6\end{bmatrix},then\;verify \;that:(i)\quad(A')'=A$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}1 & 0 & -1\\2 & 1 & 3\\0 & 1 & 1\end{bmatrix},then\;verify\;that\;A^2+A=A(A+I),where\;I\;is\;3\times 3\;unit\;matrix.$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}2 & 1\end{bmatrix}\;B=\begin{bmatrix}5 & 3 & 4\\8 & 7 & 6\end{bmatrix}\;andC=\begin{bmatrix}-1 & 2 & 1\\1 & 0 & 2\end{bmatrix},verify\;that\;A(B+C)=(AB+AC)$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $\begin{bmatrix}2 & 1 & 3\end{bmatrix}\begin{bmatrix}-1 & 0 & -1\\-1 & 1 & 0\\0 & 1 & 1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=A,find\;A.$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $P=\begin{bmatrix}x & 0 &0\\0 & y & 0\\0 & 0 & z\end{bmatrix}\;$ and $\;Q=\begin{bmatrix}a & 0 & 0\\0 & b & 0\\0 & 0 & c\end{bmatrix}$ prove that $PQ=\begin{bmatrix}xa & 0 & 0\\0 & yb & 0\\0 & 0 & zc\end{bmatrix}=QP$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}1 & 2\\-2 & 1\end{bmatrix},B=\begin{bmatrix}2 & 3\\3 & -4\end{bmatrix}\;and\;C=\begin{bmatrix}1 & 0\\-1 & 0\end{bmatrix},verify:(i)\;(AB)C=A(BC)$

asked Dec 22, 2012 by sreemathi.v

1 answer

Which of the following is an example of matrices A,B and C such that AB=AC,where A is non-zero matrix,but $B\neq C.$

asked Dec 22, 2012 by sreemathi.v

1 answer

If A=[3 , 5],B=[7, 3],then find a non-zero matrix C such that AC=BC.

asked Dec 22, 2012 by sreemathi.v

1 answer

If X and Y are $2\times 2$matrices,then solve the following matrix equation for X and Y\[2X+3Y=\begin{bmatrix}2 & 3\\4 & 0\end{bmatrix},3X+2Y=\begin{bmatrix}-2 & 2\\1 & -5\end{bmatrix}.\]

asked Dec 22, 2012 by sreemathi.v

1 answer

Solve for x and y:\[x\begin{bmatrix}2\\1\end{bmatrix}+y\begin{bmatrix}3\\5\end{bmatrix}+\begin{bmatrix}-8\\-11\end{bmatrix}=0\]

asked Dec 22, 2012 by sreemathi.v

1 answer

Given $A=\begin{bmatrix}2 & 4 & 0\\3 & 9 & 6\end{bmatrix}\;and\;B=\begin{bmatrix}1& 4\\2 & 8\\1 & 3\end{bmatrix}.\;Is \;(AB)'=B'A'?$

asked Dec 22, 2012 by sreemathi.v

1 answer

Show by an example that for $A\neq 0,B\neq 0,AB=0.$

asked Dec 22, 2012 by sreemathi.v

1 answer

If possible,find BA and AB,where\[A=\begin{bmatrix}2 & 1 & 2\\1 & 2 & 4\end{bmatrix},B=\begin{bmatrix}4 & 1\\2 & 3\\1 & 2\end{bmatrix}\]

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}3 & -4\\1 & 1\\2 & 0\end{bmatrix}\;and\;B=\begin{bmatrix}2 & 1 & 2\\1 & 2 & 4\end{bmatrix} \;$ then verify that $(BA)^2\neq B^2A^2$

asked Dec 22, 2012 by sreemathi.v

1 answer

Find A,if $\begin{bmatrix}4\\1\\3\end{bmatrix}A=\begin{bmatrix}4 & 8 & 4\\1 & 2 &1\\3 & 6 & 3\end{bmatrix}$

asked Dec 22, 2012 by sreemathi.v

1 answer

Find the matrix A satisfying the matrix equation $\begin{bmatrix}2 & 1\\3 & 2\end{bmatrix}A\begin{bmatrix}3 & 2\\5 & 3\end{bmatrix}=\begin{bmatrix}1 & 0\\0 & 12\end{bmatrix}$

asked Dec 22, 2012 by sreemathi.v

1 answer

Does $A=\begin{bmatrix}5 & 3\\1 & 2\end{bmatrix}$ satisfy the equation $\;A^2-3A-7I=0$ and hence find $\;A^{-1}$

asked Dec 22, 2012 by sreemathi.v

1 answer

Find the value of $x$ if $\begin{bmatrix}1 & x &1\end{bmatrix}\begin{bmatrix}1 & 3 & 2\\2 & 5 & 1\\15 & 3 & 2\end{bmatrix}\begin{bmatrix}1\\2\\x\end{bmatrix}=0$.

asked Dec 22, 2012 by sreemathi.v

1 answer

If $A=\begin{bmatrix}0 & 1\\1& 1\end{bmatrix}\;and\;B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix},\;show\;that\;(A+B)(A-B)\neq A^2-B^2$

asked Dec 22, 2012 by sreemathi.v

1 answer

Find non-zero values of $x$ satisfying the matrix equation $x\begin{bmatrix}2x & 2\\3 & x\end{bmatrix}+2\begin{bmatrix}8 & 5x\\4 & 4x\end{bmatrix}=2\begin{bmatrix}(x^2+8) & 24\\(10) & 6x\end{bmatrix}$

asked Dec 22, 2012 by sreemathi.v

1 answer

If $X=\begin{bmatrix}3 & 1 & 1\\5 & 2 & 3\end{bmatrix}$ and $Y=\begin{bmatrix}2 & 1 & 1\\7 & 2 & 4\end{bmatrix}$, find $X + Y$

asked Dec 22, 2012 by sreemathi.v

1 answer

If possible,find the sum of the matrices A and B,where $A=\begin{bmatrix}\sqrt 3 & 1\\2 & 3\end{bmatrix},and\; B=\begin{bmatrix}x & y & z\\a & b &6\end{bmatrix}$

asked Dec 22, 2012 by sreemathi.v

1 answer

Find values of a and b if A=B,where A=$\begin{bmatrix}a + 4 & 3b\\8 & - 6\end{bmatrix},B=\begin{bmatrix}2a + 2 & b^2 + 2\\8 & b^2 - 5b\end{bmatrix}$

asked Dec 22, 2012 by sreemathi.v

1 answer

Construct a $3\times 2$ matrix whose elements are given by $a_{ij}=e^{i.x}\sin jx$

asked Dec 22, 2012 by sreemathi.v

1 answer

Construct $a_{2\times 2}$ matrix where $a_{ij} =\frac{(i + 2j)^2}{2}$

asked Dec 22, 2012 by sreemathi.v

1 answer

What is the order of the matrix $A=\begin{bmatrix} a & 1 & x \\ 2 & \sqrt 3 & x^2 y \\ 0 & 5 & \frac{2}{5} \end{bmatrix}$

asked Dec 21, 2012 by sreemathi.v

1 answer

If a matrix has 28 elements, what are the possible orders it can have?

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False: The principal value of $\sin^{-1}\begin{bmatrix}\cos\bigg(\sin^{-1}\frac{1}{2}\bigg)\end{bmatrix}\;is\;\Large {\frac{\pi}{3}}$

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False: The minimum value of n for which $\tan^{-1}\Large {\frac{n}{\pi}}\;>\Large {\frac{\pi}{4}},\normalsize\;n\in N,$ is valid is 5

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False:The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging x and y axes.

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False: The least numerical value, either positive or negative of angle $\theta$ is called principal value of the inverse trigonometric function.

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False: The domain of trigonometric functions can be restricted to any one of their branch(not necessarily principal value) in order to obtain their inverse function.

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False: The value of the expression $(\cos^{-1}x)^2$ is equal to $\sec^2x.$

asked Dec 21, 2012 by sreemathi.v

1 answer

True or False: All trigonometric functions have inverse over their respective domains.

asked Dec 21, 2012 by sreemathi.v

1 answer

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