Let A- - 1 0 0 2 1 0 3 2 1 - and U1- U2- U3 be columnmatrices satisfying AU1- - 1 0 0 - AU2- - 2 3 0 - AU3- - 2 3 1 - If U is 3- 3 matrix whose columns are U1- U2- U3- then - U -

The correct option is C 3 A=[ 1 amp; 0 amp; 0; 2 amp; 1 amp; 0; 3 amp; 2 amp; 1 ] Let U_1=[ a_ 1; b_ 1; c_ 1 ] , U_2=[ a_ 2; ...

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Byju's Answer

Standard XII

Mathematics

Addition and Subtraction of a Matrix

Let A= [ 1 ...

Question

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

3

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\frac{3}{2}

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Solution

The correct option is C $3$
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦$
Let $U_{1} = ⎡ ⎢ ⎣ \begin{matrix} a_{1} b_{1} c_{1} \end{matrix} ⎤ ⎥ ⎦$ , $U_{2} = ⎡ ⎢ ⎣ \begin{matrix} a_{2} b_{2} c_{2} \end{matrix} ⎤ ⎥ ⎦$ and $U_{3} = ⎡ ⎢ ⎣ \begin{matrix} a_{3} b_{3} c_{3} \end{matrix} ⎤ ⎥ ⎦$
Given $A U_{1} = ⎡ ⎢ ⎣ \begin{matrix} 100 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow a_{1} = 1; 2 a_{1} + b_{1} = 0; 3 a_{1} + 2 b_{1} + c_{1} = 0$
simplifying gives $a_{1} = 1; b_{1} = - 2; c_{1} = 1$
$A U_{2} = ⎡ ⎢ ⎣ \begin{matrix} 230 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow a_{2} = 2; 2 a_{2} + b_{2} = 3; 3 a_{2} + 2 b_{2} + c_{2} = 0$
simplifying gives $a_{2} = 2; b_{2} = - 1; c_{2} = - 4$
$A U_{3} = ⎡ ⎢ ⎣ \begin{matrix} 231 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow a_{3} = 2; 2 a_{3} + b_{3} = 3; 3 a_{3} + 2 b_{3} + c_{3} = 1$
simplifying gives $a_{3} = 2; b_{3} = - 1; c_{3} = - 3$
$∴ U = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 2 - 2 & - 1 & - 1 1 & - 4 & - 3 \end{matrix} ⎤ ⎥ ⎦$
$| U | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 2 - 2 & - 1 & - 1 1 & - 4 & - 3 \end{matrix} ∣ ∣ ∣ ∣ = 3$
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

| 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} ⎤ ⎥ ⎦

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. 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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Let A=⎡⎢⎣100210321⎤⎥⎦ and U1,U2,U3 be columnmatrices satisfying AU1=⎡⎢⎣100⎤⎥⎦,AU2=⎡⎢⎣230⎤⎥⎦,AU3=⎡⎢⎣231⎤⎥⎦. If U is3×3 matrix whose columns are U1,U2,U3, then |U|=