A-1 0 0 2 1 0 3 2 1- U1- U2 and U3 are columns 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- U3 then answer the following questionThe value of -U- is

The correct option is A 3 Given, #160; AU_1=[1 0 0] #160; ⇒ U_1=A^ 1[1 0 0] #160; #160;...(1) AU_2 = #160;[2 ...

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Byju's Answer

Standard XII

Mathematics

Definition of a Determinant

A=[1 0 0 2...

Question

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦, U_{1}, U_{2}$ 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 $U$ is $3 \times 3$ matrix whose columns are $U_{1}, U_{2}, U_{3}$ then answer the following question
The value of $| U |$ is

3

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\frac{3}{2}

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2

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Solution

The correct option is A $3$
Given,
$A U_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$ $\Rightarrow U_{1} = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$ ...(1)
$A U_{2} = ⎡ ⎢ ⎣ \begin{matrix} 230 \end{matrix} ⎤ ⎥ ⎦$ $\Rightarrow U_{2} = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 230 \end{matrix} ⎤ ⎥ ⎦$ ...(2)
$A U_{3} = ⎡ ⎢ ⎣ \begin{matrix} 231 \end{matrix} ⎤ ⎥ ⎦$ $\Rightarrow U_{3} = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 231 \end{matrix} ⎤ ⎥ ⎦$ ...(3)
Also, given
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦$
$| A | = 1$
Now,
$a d j A = C^{T} = {⎡ ⎢ ⎣ \begin{matrix} 1 & - 2 & 1 0 & 1 & - 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦}^{T}$
$\Rightarrow a d j A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow A^{- 1} = \frac{a d j A}{| A |} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦$
Put the value of $A^{- 1}$ in (1), (2) and (3)
$U_{1} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$ $\Rightarrow U_{1} = ⎡ ⎢ ⎣ \begin{matrix} 1 - 2 1 \end{matrix} ⎤ ⎥ ⎦$
$U_{2} = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 - 2 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 230 \end{matrix} ⎤ ⎥ ⎦$ $\Rightarrow U_{2} = ⎡ ⎢ ⎣ \begin{matrix} 2 - 1 - 4 \end{matrix} ⎤ ⎥ ⎦$
$U_{3} = A^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 231 \end{matrix} ⎤ ⎥ ⎦$ $\Rightarrow U_{3} = ⎡ ⎢ ⎣ \begin{matrix} 2 - 1 - 3 \end{matrix} ⎤ ⎥ ⎦$
$∴ U = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 2 - 2 & - 1 & - 1 1 & - 4 & - 3 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow | U | = - 1 - 14 + 18 = 3$

Hence, option A.

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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 value of

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

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}

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 | =

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}

Q. Let the matrix

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

satisfy

M^{n} = M^{n - 2} + M^{2} - I

for

n = 3, 4, 5, 6, \dots

where

I

is the identity matrix of order

3

.
Also

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

are column matrices satisfying

M^{50} U_{1} = ⎡ ⎢ ⎣ \begin{matrix} 12525 \end{matrix} ⎤ ⎥ ⎦, M^{50} U_{2} = ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦, M^{50} U_{3} = ⎡ ⎢ ⎣ \begin{matrix} 001 \end{matrix} ⎤ ⎥ ⎦

and

U

is a

3 \times 3

matrix whose columns are

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

.
Then

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A=⎡⎢⎣100210321⎤⎥⎦,U1,U2 and U3 are columns matrices satisfying AU1=⎡⎢⎣100⎤⎥⎦,AU2=⎡⎢⎣230⎤⎥⎦,AU3=⎡⎢⎣231⎤⎥⎦ and U is 3×3 matrix whose columns are U1,U2,U3 then answer the following questionThe value of |U| is