For a matrixA - beginpmatrix 1 0 0 - 12 1 0 - 13 2 1lendpmatrix - if U 1- U 2 and U 3 are 3 - 1 column matrices satisfying AU - 1-egin pmatrix 1 0 0end pmatrix - - AU - 2- Weginpmatrix 2113 - 0lendpmatrix - AU - 3- beginpmatrix 2113 - 1endpmatrix and U is 3 - 3 matrix whose columns are U 1- U 2 and U 3 - Then sum of the elements of U -1 isA- 6B- 0 zeroC- 1D- 2-3

The correct option is B 0 (zero)Let U_i= nbsp;[ a_i; b_i; c_i nbsp; nbsp; ], i = 1, 2, 3 AU_1 = nbsp;[ a_1; 2a_1 + b_1; 3a_1 + 2 ...

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Byju's Answer

Standard XII

Mathematics

Identity Matrix

For a matrixA...

Question

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 $U$ is $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$ and $U_{3} .$ Then sum of the elements of $U^{- 1}$ is

6

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0

(zero)

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1

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\frac{2}{3}

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Solution

The correct option is B $0$ (zero)
Let $U_{i} = ⎛ ⎜ ⎝ \begin{matrix} a_{i} b_{i} c_{i} \end{matrix} ⎞ ⎟ ⎠, i = 1, 2, 3$
$A U_{1} = ⎛ ⎜ ⎝ \begin{matrix} a_{1} 2 a_{1} + b_{1} 3 a_{1} + 2 b_{1} + c_{1} \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 100 \end{matrix} ⎞ ⎟ ⎠$

$\Rightarrow a_{1} = 1, b_{1} = - 2, c_{1} = 1$

Similarly,
$A U_{2} = ⎛ ⎜ ⎝ \begin{matrix} a_{2} 2 a_{2} + b_{2} 3 a_{2} + 2 b_{2} + c_{2} \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 230 \end{matrix} ⎞ ⎟ ⎠$

$\Rightarrow a_{2} = 2, b_{2} = - 1, c_{2} = - 4$

Similarly,
$A U_{3} = ⎛ ⎜ ⎝ \begin{matrix} a_{3} 2 a_{3} + b_{3} 3 a_{3} + 2 b_{3} + c_{3} \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 231 \end{matrix} ⎞ ⎟ ⎠$

$\Rightarrow a_{3} = 2, b_{3} = - 1, c_{3} = - 3$

Therefore, the matrix $U$ is
$U = ⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 2 - 2 & - 1 & - 1 1 & - 4 & - 3 \end{matrix} ⎞ ⎟ ⎠$

$U^{- 1} = \frac{1}{3} ⎛ ⎜ ⎝ \begin{matrix} - 1 & - 2 & 0 - 7 & - 5 & - 3 9 & 6 & 3 \end{matrix} ⎞ ⎟ ⎠$
The sum of the element of $U^{- 1}$ is zero.

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Similar questions

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}

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 value of

| U |

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 value of

[\begin{matrix} 3 & 2 & 0 \end{matrix}] U ⎡ ⎢ ⎣ \begin{matrix} 320 \end{matrix} ⎤ ⎥ ⎦

Q. Let

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

and

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

be 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} ⎤ ⎥ ⎦

. If U is

3 \times 3

matrix whose columns are

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

, then

| U | =

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For a matrix A=⎛⎜⎝100210321⎞⎟⎠, if U1,U2 and U3 are 3×1 column matrices satisfying AU1=⎛⎜⎝100⎞⎟⎠, AU2=⎛⎜⎝230⎞⎟⎠,AU3=⎛⎜⎝231⎞⎟⎠ and U is 3×3 matrix whose columns are U1,U2 and U3. Then sum of the elements of U−1 is