Let A -1-102 13121 and B -123213011- Find AT- BT and verify thati A-BT-AT-BTii A BT-BT ATiii 2 AT-2 AT-

Given: #160;A=1 10213121 #160;and #160;B=123213011AT=121 112031 #160;and #160; #160;BT=120211331i #160;A+B=1 10213121+123213011 #160; #8658;A+ ...

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Byju's Answer

Standard XII

Mathematics

Associative Law of Binary Operation

Let A =1-102 ...

Question

Let $A = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{matrix}] .$ Find A^T, B^T and verify that
(i) (A + B)^T = A^T + B^T
(ii) (AB)^T = B^T A^T
(iii) (2A)^T = 2A^T.

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Solution

$Given : A = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{matrix}] A^{T} = [\begin{matrix} 1 & 2 & 1 \\ - 1 & 1 & 2 \\ 0 & 3 & 1 \end{matrix}] and B^{T} = [\begin{matrix} 1 & 2 & 0 \\ 2 & 1 & 1 \\ 3 & 3 & 1 \end{matrix}] (i) A + B = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] + [\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{matrix}] \Rightarrow A + B = [\begin{matrix} 1 + 1 & - 1 + 2 & 0 + 3 \\ 2 + 2 & 1 + 1 & 3 + 3 \\ 1 + 0 & 2 + 1 & 1 + 1 \end{matrix}] \Rightarrow A + B = [\begin{matrix} 2 & 1 & 3 \\ 4 & 2 & 6 \\ 1 & 3 & 2 \end{matrix}] \Rightarrow {(A + B)}^{T} = [\begin{matrix} 2 & 4 & 1 \\ 1 & 2 & 3 \\ 3 & 6 & 2 \end{matrix}] . . . (1) Now, A^{T} + B^{T} = [\begin{matrix} 1 & 2 & 1 \\ - 1 & 1 & 2 \\ 0 & 3 & 1 \end{matrix}] + [\begin{matrix} 1 & 2 & 0 \\ 2 & 1 & 1 \\ 3 & 3 & 1 \end{matrix}] \Rightarrow A^{T} + B^{T} = [\begin{matrix} 1 + 1 & 2 + 2 & 1 + 0 \\ - 1 + 2 & 1 + 1 & 2 + 1 \\ 0 + 3 & 3 + 3 & 1 + 1 \end{matrix}] \Rightarrow A^{T} + B^{T} = [\begin{matrix} 2 & 4 & 1 \\ 1 & 2 & 3 \\ 3 & 6 & 2 \end{matrix}] . . . (2) \Rightarrow {(A + B)}^{T} = A^{T} + B^{T} [From eqs . (1) and (2)] (i i) A B = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{matrix}] \Rightarrow A B = [\begin{matrix} 1 - 2 + 0 & 2 - 1 + 0 & 3 - 3 + 0 \\ 2 + 2 + 0 & 4 + 1 + 3 & 6 + 3 + 3 \\ 1 + 4 + 0 & 2 + 2 + 1 & 3 + 6 + 1 \end{matrix}] \Rightarrow A B = [\begin{matrix} - 1 & 1 & 0 \\ 4 & 8 & 12 \\ 5 & 5 & 10 \end{matrix}] \Rightarrow {(A B)}^{T} = [\begin{matrix} - 1 & 4 & 5 \\ 1 & 8 & 5 \\ 0 & 12 & 10 \end{matrix}] . . . (1) Now, B^{T} A^{T} = [\begin{matrix} 1 & 2 & 0 \\ 2 & 1 & 1 \\ 3 & 3 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 1 \\ - 1 & 1 & 2 \\ 0 & 3 & 1 \end{matrix}] \Rightarrow B^{T} A^{T} = [\begin{matrix} 1 - 2 + 0 & 2 + 2 + 0 & 1 + 4 + 0 \\ 2 - 1 + 0 & 4 + 1 + 3 & 2 + 2 + 1 \\ 3 - 3 + 0 & 6 + 3 + 3 & 3 + 6 + 1 \end{matrix}] \Rightarrow B^{T} A^{T} = [\begin{matrix} - 1 & 4 & 5 \\ 1 & 8 & 5 \\ 0 & 12 & 10 \end{matrix}] . . . (2) \Rightarrow {(A B)}^{T} = B^{T} A^{T} [From eqs . (1) and (2)] (iii) 2 A = 2 [\begin{matrix} 1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] \Rightarrow 2 A = [\begin{matrix} 2 & - 2 & 0 \\ 4 & 2 & 6 \\ 2 & 4 & 2 \end{matrix}] \Rightarrow {(2 A)}^{T} = [\begin{matrix} 2 & 4 & 2 \\ - 2 & 2 & 4 \\ 0 & 6 & 2 \end{matrix}] . . . (1) Now, 2 A^{T} = 2 [\begin{matrix} 1 & 2 & 1 \\ - 1 & 1 & 2 \\ 0 & 3 & 1 \end{matrix}] \Rightarrow 2 A^{T} = [\begin{matrix} 2 & 4 & 2 \\ - 2 & 2 & 4 \\ 0 & 6 & 2 \end{matrix}] . . . (2) \Rightarrow {(2 A)}^{T} = 2 A^{T} [From eqs . (1) and (2)]$

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Q. Let

A = [\begin{matrix} 2 & - 3 \\ - 7 & 5 \end{matrix}] and B = [\begin{matrix} 1 & 0 \\ 2 & - 4 \end{matrix}]

, verify that
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(iii) (A − B)^T = A^T − B^T
(iv) (AB)^T = B^T A^T

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Q. Find the distance between the points

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If $A = [\begin{matrix} 12 - 13 \end{matrix}] B = [\begin{matrix} 4015 \end{matrix}] C = [\begin{matrix} 20 1 - 2 \end{matrix}]$ ,
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(ii) A (BC) = (AB) C

(iii) (a + b)B = aB + bB

(iv) a (C - A) = aC - aA

(v) $(A^{T})^{T}$ = A

(vi) $(b A)^{T}$ = b $A^{T}$

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(ix) $(A - B)^{T} = A^{T} - B^{T}$

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Let A=1-102 131 21 and B=123213011. Find AT, BT and verify that (i) (A + B)T = AT + BT (ii) (AB)T = BT AT (iii) (2A)T = 2AT.

Let $A = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{matrix}] .$ Find A^T, B^T and verify that
(i) (A + B)^T = A^T + B^T
(ii) (AB)^T = B^T A^T
(iii) (2A)^T = 2A^T.