If A - begin bmatrix 2 - 2 -4 1-4 - 2 -4 12 -1 - 5end bmatrix and B - bmatrix 1 -1 - 0 1283 - 410 - 1 - 2lend bmatrix - then find BA and use this to solve the system of equations y-2 z-7- x-y-3 and 2 x-3 y-4 z-17

We have, A = [ 2 amp; 2 amp; 4; 4 amp; 2 amp; 4; 2 amp; 1 amp; 5 ] and B = [ 1 amp; 1 amp; 0; 2 amp; 3 amp; 4; 0 ...

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Byju's Answer

Standard XII

Mathematics

Multiplication of Matrices

If A = begin ...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ , then find BA and use this to solve the system of equations y + 2z = 7, x - y = 3 and 2x + 3y + 4z = 17.

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Solution

We have, $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$
$∴ B A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 6 & 0 & 0 0 & 6 & 0 0 & 0 & 6 \end{matrix} ⎤ ⎥ ⎦ = 6 I$
$∴ B^{- 1} = \frac{A}{6} = \frac{1}{6} A = \frac{1}{6} ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦$ .....(i)
Also, x - y = 3, 2x+3y+ 4z = 17 and y + 2z = 7
$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3177 \end{matrix} ⎤ ⎥ ⎦$
$∴ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = {⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦}^{- 1} ⎡ ⎢ ⎣ \begin{matrix} 3177 \end{matrix} ⎤ ⎥ ⎦ = \frac{1}{6} ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 3177 \end{matrix} ⎤ ⎥ ⎦ [using Eq. (i)]$
$= \frac{1}{6} ⎡ ⎢ ⎣ \begin{matrix} 6 + 34 - 28 - 12 + 34 - 28 6 - 17 + 35 \end{matrix} ⎤ ⎥ ⎦ = \frac{1}{6} ⎡ ⎢ ⎣ \begin{matrix} 12 - 6 24 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 2 - 1 4 \end{matrix} ⎤ ⎥ ⎦$
$∴$ x = 2, y = -1 and z = 4

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Similar questions

If $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 3 & 5 3 & 2 & - 4 1 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎦$ , find $A^{- 1}$ . Using $A^{- 1}$ solve the system of equations $2 x - 3 y + 5 z = 11, 3 x + 2 y - 4 z = - 5, x + y - 2 z = - 3$

Q. If

A =

⎡ ⎢ ⎣ \begin{matrix} 2 & - 3 & 5 3 & 2 & - 4 1 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

, then find

A^{- 1}

.
Use it to solve the system of equations

2 x - 3 y + 5 z = 11 3 x + 2 y - 4 z = - 5 x + y - 2 z = - 3

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 3 & 5 3 & 2 & - 4 1 & 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

, find

A^{- 1}

. Use it to solve the system of equations.

2 x - 3 y + 5 z = 11

3 x + 2 y - 4 z = - 5

x + y - 2 z = - 3

Q. Determine the product

⎡ ⎢ ⎣ \begin{matrix} - 4 & 4 & 4 - 7 & 1 & 3 5 & - 3 & - 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 1 & - 2 & - 2 2 & 1 & 3 \end{matrix} ⎤ ⎥ ⎦

and use it to solve the system of equations

x - y + z = 4, x - 2 y - 2 z = 9, 2 x + y + 3 z = 1

Q. Given two matrices

A

and

B

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 2 & 3 1 & 4 & 1 1 & - 3 & 2 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} 11 & - 5 & - 14 - 1 & - 1 & 2 - 7 & 1 & 6 \end{matrix} ⎤ ⎥ ⎦

Find

A B

and use this result to solve the following system of equations:

x - 2 y + 3 z = 6, x + 4 y + z = 12, x - 3 y + 2 z = 1

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If A=⎡⎢⎣22−4−42−42−15⎤⎥⎦ and B=⎡⎢⎣1−10234012⎤⎥⎦, then find BA and use this to solve the system of equations y + 2z = 7, x - y = 3 and 2x + 3y + 4z = 17.

If $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$ , then find BA and use this to solve the system of equations y + 2z = 7, x - y = 3 and 2x + 3y + 4z = 17.