Let A- 1 0 0- 2 1 0- 3 2 1- - If u1 and u2 are column matrices such that Au1- 1- 0- 0- - and Au2- 0- 1- 0- - then u1-u2 is equal to -

Given that, Au_1=[ 1; 0; 0; ] ∴ u_1=A^ 1[ 1; 0; 0; ] nbsp; nbsp;∵ [AB=B^ 1A^ 1] Also, Au_2=[ 0; 1; 0; ] ∴ u_2=A^ 1[ 0; 1; 0; ] nbsp; we know that ...

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Byju's Answer

Standard XII

Mathematics

Scalar Matrix

Let A=[ 1 0 0...

Question

Let $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦$ . If $u_{1}$ and $u_{2}$ are column matrices such that $A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$ and $A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦$ , then $u_{1} + u_{2}$ is equal to :

⎡ ⎢ ⎣ \begin{matrix} - 1 10 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} - 1 1 - 1 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} - 1 - 1 0 \end{matrix} ⎤ ⎥ ⎦

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⎡ ⎢ ⎣ \begin{matrix} 1 - 1 - 1 \end{matrix} ⎤ ⎥ ⎦

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Solution

The correct option is D $⎡ ⎢ ⎣ \begin{matrix} 1 - 1 - 1 \end{matrix} ⎤ ⎥ ⎦$
Given that,
$A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$
$∴ u_{1} = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$ $∵ [A B = B^{- 1} A^{- 1}]$
Also,
$A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦$
$∴ u_{2} = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦$
we know that,
$A^{- 1} = \frac{1}{| A |} a d j (A)$
$| A | = 1$
Minor of matrix A is $⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 0 & 1 & 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
Cofactor of matrix A is $⎡ ⎢ ⎣ \begin{matrix} 1 & - 2 & 1 0 & 1 & - 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$A d j (A) = [Cofactor of matrix A]^{T}$
$A d j (A) = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ A^{- 1} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦ \Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 - 2 1 \end{matrix} ⎤ ⎥ ⎦$
also,
$u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦ \Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 01 - 2 \end{matrix} ⎤ ⎥ ⎦$
$∴ u_{1} + u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 - 1 \end{matrix} ⎤ ⎥ ⎦$ .

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Similar questions

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

u_{1}

and

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A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦

and

A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦

, then

u_{1} + u_{2}

is equal to

Q.
Let A

= ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

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A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦

and

A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦

then

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Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

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and

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Let A = .

If u₁ and u₂ are column matrices such that Au₁= and Au₂= ,

then u₁+u₂is equal to

Q. For a matrix

A ⎛ ⎜ ⎝ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎞ ⎟ ⎠

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A U_{1} = ⎛ ⎜ ⎝ \begin{matrix} 100 \end{matrix} ⎞ ⎟ ⎠, A U_{2} ⎛ ⎜ ⎝ \begin{matrix} 230 \end{matrix} ⎞ ⎟ ⎠, A U_{3} = ⎛ ⎜ ⎝ \begin{matrix} 231 \end{matrix} ⎞ ⎟ ⎠

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Let A=⎡⎢⎣100210321⎤⎥⎦. If u1 and u2 are column matrices such that Au1=⎡⎢⎣100⎤⎥⎦ and Au2=⎡⎢⎣010⎤⎥⎦, then u1+u2 is equal to :