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 B [ 1; 1; 1 ] Given: A=[ 1 amp; 0 amp; 0; 2 amp; 1 amp; 0; 3 amp; 2 amp; 1 ], Au_1=[ 1; 0; 0 ], Au_2=[ 0; 1; 0 ...

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Byju's Answer

Standard X

Mathematics

Equality of Matrices

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 B $⎡ ⎢ ⎣ \begin{matrix} 1 - 1 - 1 \end{matrix} ⎤ ⎥ ⎦$
Given: $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦, A u_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦, A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦$

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

sol: $A u_{1} + A u_{2} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 110 \end{matrix} ⎤ ⎥ ⎦$

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

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

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

$A^{- 1} = \frac{A d j (A)}{d e t (A)}$

$d e t (A) = 1$

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

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

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

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Similar questions

Q.
Let A

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

.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

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} ⎞ ⎟ ⎠

, 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