Matrix

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

Click on each question to see answer & solution.

Question No: #51

If `A`, `B`, `C` are 3 given matrices such that $ A = \begin{pmatrix} 3 & 5 \\ 2 & a \end{pmatrix}$, $ B = \begin{pmatrix} 4 & b \\ 2 & 9 \end{pmatrix}$ and $ C = \begin{pmatrix} 22 & 14 \\ a & -1 \end{pmatrix}$, find `a` and `b` such that `A\cdot\B = C^T`.

Question No: #52

Find `A^100` if $ A = \begin{pmatrix} \omega & 0 \\ 0 & \omega \end{pmatrix} $ , where `\omega` is the imaginary cube root of 1.
Ans: `A`
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Question No: #53

If $ A = \begin{pmatrix} i & - i \\ - i & i \end{pmatrix} $ and $ B = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} $ then show that `A^8=128\ B`

Question No: #54

If $ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} $ and `n in mathbb{N}`, then show that `A^n=2^{n-1}A`
Ans: N.A.
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Question No: #55

If $ A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} $, then show that $ A^n = \begin{pmatrix} 1+2n & -4n \\ n & 1-2n \end{pmatrix} $ where `n \in \mathbb{N}`
Ans: N.A.
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Question No: #56

If $ A = \begin{bmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{bmatrix} $, then find the matrix `A^50`
Ans: $ \begin{bmatrix} 1 & 0 \\ 25 & 1 \end{bmatrix} $
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Question No: #57

If $ A = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix} $ then prove that `A^2-2A+I_2=0`. Hence find the matrix `A^50`
Ans: $ \begin{bmatrix} 1 & 0 \\ -50 & 1 \end{bmatrix} $
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Question No: #58

If $ A = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} $ then find `A\A^T`

Question No: #59

If $ A = \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} $ then find `A\A^T`

Question No: #60

If $ P = \begin{pmatrix} 1 & 2 & 1 \\ 1 & 3 & 1 \end{pmatrix} $ and `Q = PP^T`, then find the matrix `Q`