Matrix

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

Click on each question to see answer & solution.

Question No: #61

If $ A = \begin{pmatrix} -2 & 1 & 3 \\ 0 & 4 & -1 \end{pmatrix} $ and $ B = \begin{pmatrix} 2 & 1 \\ -3 & 0 \\ 4 & -5 \end{pmatrix} $, then prove that `(AB)^T = B^TA^T`

Question No: #62

If $ A = \begin{bmatrix} 1 & 2 & 3 & 4 \end{bmatrix} $ and $ B = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} $ then find `AB` and `BA`

Question No: #63

Express the following matrix (`A`) as the sum of symmetric & skew-symmetric matrix. $ A = \begin{bmatrix} -3 & 4 & 1 \\ 2 & 3 & 0 \\ 1 & 4 & 5 \end{bmatrix}$
Ans: $ \begin{bmatrix} -3 & 3 & 1 \\ 3 & 3 & 2 \\ 1 & 2 & 5 \end{bmatrix} + \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & -2 \\ 0 & 2 & 0 \end{bmatrix}$
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Question No: #64

Express the following matrix (`A`) as the sum of symmetric & skew-symmetric matrix. $ A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$

Question No: #65

If $ A = \begin{bmatrix} 1 & x \\ x^2 & 4y \end{bmatrix} $, $ B = \begin{bmatrix} -3 & 1 \\ 1 & 0 \end{bmatrix} $ and $ adj(A) + B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $ then find the value of `x` and `y`.

Question No: #66

If $ A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ then show that `A^2-5A-2I_2=O` where $ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $ O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$. Hence find `A^{-1}`
Ans: $ A^{-1} = \frac{1}{2} \begin{pmatrix} -4 & 3 \\ 2 & -1 \end{pmatrix}$
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Question No: #67

If $ A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix} $ and `a`, `b` are two real numbers such that `A^2+aA+bI=O` (where `I` & `O` are unit & zero matrix), then find `A^{-1}`
Ans: $ A^{-1} = \begin{bmatrix} 7 & 2 \\ 1 & 5 \end{bmatrix} $
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Question No: #68

If $ A = \begin{pmatrix} 2x & 0 \\ x & x \end{pmatrix} $ and $ A^{-1} = \begin{pmatrix} 1 & 0 \\ -1 & 2 \end{pmatrix} $, then find the value of `x`

Question No: #69

If $ A = \begin{pmatrix} 2 & -3 \\ -4 & 7 \end{pmatrix} $ and `2A^{-1}=kI-A` where `I` is unit matrix. Find the value of `k`.

Question No: #70

If $A= \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ then show that `6A^{-1}+5I=A`
Ans: N.A.
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