For a matrix A - 1 0 0- 2 1 0- 3 2 1 - if U1- U2 and U3 are 3- 1 column matrices satisfying AU1 - - 1- 0- 0 - AU2- 2- 3- 0 - AU3 - - 2- 3- 1 - and U is 3- 3 matrix whose columns are U1- U2 and U3Then sum of the elements of U-1 is

The correct option is A 0 (zero) Let U_i = [ a_i; b_i #160;; c_i #160; ] i = 1, 2, 3 AU_1 = #160;[ a_1; 2a_1 + b_1 ...

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Standard XII

Mathematics

Finding Inverse Using Elementary Transformations

For a matrix ...

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 (z e r o)

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Solution

The correct option is A $0 (z e r o)$
Let $U_{i} = ⎛ ⎜ ⎝ \begin{matrix} a_{i} b_{i} c_{i} \end{matrix} ⎞ ⎟ ⎠ i = 1, 2, 3$
$A U_{1} = ⎛ ⎜ ⎝ \begin{matrix} a_{1} 2 a_{1} + b_{1} 3 a_{1} + 2 b_{1} + c_{1} \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 100 \end{matrix} ⎞ ⎟ ⎠$ , So $a_{1} = 1, b_{1} = - 2, c_{1} = 1$
$A U_{2} = ⎛ ⎜ ⎝ \begin{matrix} a_{2} 2 a_{2} + b^{2} 3 a_{2} + 2 b_{2} + c_{2} \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} 230 \end{matrix} ⎞ ⎟ ⎠$ ,
So, $a_{2} = 2, b_{2} = - 1, c_{2} = - 4$ . Similarly, $a_{3} = 2, b_{3} = - 1, c_{3} = - 3$
So, $U = ⎛ ⎜ ⎝ \begin{matrix} 1 & 2 & 2 - 2 & - 1 & - 1 1 & - 4 & - 3 \end{matrix} ⎞ ⎟ ⎠$ . So, sum of elements of $U^{- 1}$ is zero.

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Similar questions

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 |

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 | =

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} ⎤ ⎥ ⎦

. 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

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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 U3Then sum of the elements of U−1 is