Matrix

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #91

Show that matrix $A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$ satisfies the equation `A^2-4A-5I_3=O`. Hence find `A^{-1}`

Question No: #92

If $ f(x) = \begin{bmatrix} cos\ x & - sin\ x & 0 \\ sin\ x & cos\ x & 0 \\ 0 & 0 & 1 \end{bmatrix} $, then prove that `f(\alpha)\cdot\f(\beta) = f(\alpha+\beta)`.

Question No: #93

If $ A = \begin{bmatrix} 0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0 \end{bmatrix} $ and `I` is the identity matrix of order 2, then show that `(I+A) = (I-A)`$ \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} $ (or) `(I+A)(I-A)^{-1] = `$ \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix} $
Ans: N.A.
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Question No: #94

If `\alpha - \beta = (2n+1)\frac{\pi}{2}` where `n in \mathbb{Z}` then show that product of $ \begin{pmatrix} \cos^2\alpha & \cos\alpha\sin\alpha \\ \cos\alpha\sin\alpha & \sin^2\alpha \end{pmatrix} $ and $ \begin{pmatrix} \cos^2\beta& \cos\beta\sin\beta \\ \cos\beta\sin\beta & \sin^2\beta \end{pmatrix} $ is a zero matrix.
Ans: N.A.
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Question No: #95

If $ A = \begin{bmatrix} 1 & -\tan \frac{\theta}{2} \\ \tan \frac{\theta}{2} & 1 \end{bmatrix} $ and $ B = \begin{bmatrix} 1 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 1 \end{bmatrix} $ then prove that `AB^{-1} = `$ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $

Question No: #96

If $ A = \begin{pmatrix} 1 & \tan\ x \\ -\tan\ x & 1 \end{pmatrix} $ then show that $ A^T \cdot A^{-1} = \begin{pmatrix} \cos2x & -\sin2x \\ \sin2x & \cos2x \end{pmatrix} $

Question No: #97

If a matrix $ A = \begin{pmatrix} 6 & 2 & -2 \\ -2 & 2 & 2 \\ 2 & 2 & 2 \end{pmatrix} $, then show that `(A-2I)(A-4I) = 0` matrix and hence find `A^3`
Ans: $ \begin{pmatrix} 120 & 56 & -56 \\ -56 & 8 & 56 \\ 56 & 56 & 8 \end{pmatrix} $
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Question No: #98

Find the value of $ \begin{pmatrix} x & y & z \end{pmatrix} \times \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix} \times \begin{pmatrix} x \\ y \\ z \end{pmatrix} $

Question No: #99

Find the value of `p` such that $ \begin{bmatrix} 1 & p & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \times \begin{bmatrix} 1 \\ 2 \\ p \end{bmatrix} = 0 $
Ans: -2, -14
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Question No: #100

If `f(x) = x^2-5x+6` then find `f(A)` where $ A = \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} $
Ans: $ \begin{bmatrix} 1 & -1 & -3 \\ -1 & -1 & -10 \\ -5 & 4 & 4 \end{bmatrix} $
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