Matrix

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

Click on each question to see answer & solution.

Question No: #81

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

Question No: #82

If $ A = \begin{bmatrix} k & 2 \\ 2 & k \end{bmatrix} $ and `|A^3|=125`, then find value of `k`.

Question No: #83

Find the matrices `A` and `B` such that, $ A+2B = \begin{pmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{pmatrix} $ and $ 2A - B = \begin{pmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{pmatrix} $

Question No: #84

Given $ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} 2 & x & x \\ x & 4 & 5 \\ x & 6 & 7 \end{bmatrix} $. Then find the value of `x` (if possible) such that `AB = BA`
Ans: `x=0`
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Question No: #85

If $ A = \begin{pmatrix} 0 & 2b & c \\ a & b & -c \\ a & -b & c \end{pmatrix} $ and `A^TA=I`, then find the value of `a, b, c`
Ans: `a = \pm \frac{1}{\sqrt{2}}, b = \pm \frac{1}{\sqrt{6}}, c = \pm \frac{1}{\sqrt{3}}`
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Question No: #86

If $ A = \begin{bmatrix} 1 & -2 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix} $ and $ B = \begin{bmatrix} 1 & 2 & 0 \\ 2 & 3 & -1 \\ 0 & -1 & -2 \end{bmatrix} $, then show that the matrix `(A^TB)A` is a diagonal matrix.
Ans: N.A.
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Question No: #87

Show that the matrix $ A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} $ is an idempotent matrix.
Ans: N.A.
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Question No: #88

Show that the matrix $ \begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix} $ is a Nilpotent matrix of index 3.
Ans: N.A.
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Question No: #89

If $ A = \begin{bmatrix} 1 & x & -2 \\ 2 & 2 & 4 \\ 0 & 0 & 2 \end{bmatrix} $ and `A^2+2I_3=3A`, then find the value of `x`. Here `I_3` is the unit matrix of order 3.

Question No: #90

If $ P = \begin{bmatrix} -1 & 3 & 5 \\ 1 & -3 & -5 \\ -1 & 3 & -5 \end{bmatrix} $ then show that `P^2 = P` and then find a matrix `Q` such that `3P^2-2P+Q = I`.
Ans: $ \begin{bmatrix} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{bmatrix} $
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