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-

The correct option is D [[ 1; 1; 1 ]] Given:Matrices are #160; A=[ 1 amp; 0 #160; amp; 0 2 amp; 1 #160; amp; 0 3 ...

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Byju's Answer

Standard XII

Mathematics

Addition and Subtraction of a Matrix

Let A = [[ 1...

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

$A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$ and
$A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦$

To find:Matric $u_{1} + u_{2}$

Since both $A u_{1}$ and $A u_{2}$ are given, hence adding them, we get

$A u_{1} + A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦$

$A (u_{1} + u_{2}) = ⎡ ⎢ ⎣ \begin{matrix} 110 \end{matrix} ⎤ ⎥ ⎦$

Since, $A$ is a non-singular matrix,we have
$| A | \neq 0$

Hence multiplying both sides by $A^{- 1}$ from RHS we get

$A^{- 1} A (u_{1} + u_{2}) = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 110 \end{matrix} ⎤ ⎥ ⎦$

$u_{1} + u_{2} = {⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦}^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 110 \end{matrix} ⎤ ⎥ ⎦$ .......... $(1)$

Now, $| A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ∣ ∣ ∣ ∣$

$= 1 ∣ ∣ ∣ \begin{matrix} 1 & 0 2 & 1 \end{matrix} ∣ ∣ ∣ - 0 + 0$ (by expanding the determinant along row $1$ )

$\Rightarrow | A | = 1$

Now, co-factor matrix of $A$ (i.e., the matrix in which every element is replaced by corresponding co-factor)

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

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & - 2 & 1 0 & 1 & - 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ a d j (A) = {⎡ ⎢ ⎣ \begin{matrix} 1 & - 2 & 1 0 & 1 & - 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦}^{T} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$

$\Rightarrow A^{- 1} = \frac{a d j (A)}{| A |}$

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∵ | A | = 1$

From eqn $(1)$ we get

$u_{1} + u_{2} = {⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦}^{- 1} \times ⎡ ⎢ ⎣ \begin{matrix} 110 \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ \times ⎡ ⎢ ⎣ \begin{matrix} 110 \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} 1 + 0 + 0 - 2 + 1 + 0 1 - 2 + 0 \end{matrix} ⎤ ⎥ ⎦$

$∴ u_{1} + u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 - 1 \end{matrix} ⎤ ⎥ ⎦$

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Similar questions

Let A = .

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

then u₁+u₂is equal to

Q. 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

Q. Let

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

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

Q. For a matrix

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

, if

U_{1}, U_{2}

and

U_{3}

are

3 \times 1

column matrices satisfying

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

and

U

3 \times 3

matrix whose columns are

U_{1}, U_{2}

and

U_{3}

Then sum of the elements of

U^{- 1}

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

and

U_{3}

are columns matrices satisfying

A U_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦, A U_{2} = ⎡ ⎢ ⎣ \begin{matrix} 230 \end{matrix} ⎤ ⎥ ⎦, A U_{3} = ⎡ ⎢ ⎣ \begin{matrix} 231 \end{matrix} ⎤ ⎥ ⎦

and

U

3 \times 3

matrix whose columns are

U_{1}, U_{2}, U_{3}

then answer the following question

The sum of elements of

U^{- 1}

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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: