Solve the equation for x-y-z and t if 2- x z y t - -3- 1 -1- 0 2 -3- 3 5- 4 6 -

2[ x amp; z y amp; t ] +3[ 1 amp; 1; 0 amp; 2 ]=3[ 3 amp; 5; 4 amp; 6 ] ⇒[ 2x amp; 2z; 2y amp; 2t ]+[ ...

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Standard XII

Mathematics

Matrix Definition and Representation

Solve the equ...

Question

Solve the equation for $x, y, z$ and $t$ if $2 [\begin{matrix} x & z y & t \end{matrix}] + 3 [\begin{matrix} 1 & - 1 0 & 2 \end{matrix}] = 3 [\begin{matrix} 3 & 5 4 & 6 \end{matrix}]$

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Solution

$2 [\begin{matrix} x & z y & t \end{matrix}] + 3 [\begin{matrix} 1 & - 1 0 & 2 \end{matrix}] = 3 [\begin{matrix} 3 & 5 4 & 6 \end{matrix}]$
$\Rightarrow [\begin{matrix} 2 x & 2 z 2 y & 2 t \end{matrix}] + [\begin{matrix} 3 & - 3 0 & 6 \end{matrix}] = [\begin{matrix} 9 & 15 12 & 18 \end{matrix}]$
$\Rightarrow [\begin{matrix} 2 x + 3 & 2 z - 3 2 y & 2 t + 6 \end{matrix}] = [\begin{matrix} 9 & 15 12 & 18 \end{matrix}]$
Comparing the corresponding elements of these two matrices, we get:
$2 x + 3 = 9 \Rightarrow 2 x = 6 \Rightarrow x = 3$
$2 y = 12 \Rightarrow y = 6$
$2 z - 3 = 15 \Rightarrow 2 z = 18 \Rightarrow z = 9$
$2 t + 6 = 18 \Rightarrow 2 t = 12 t = 6$
$∴ x = 3, y = 6, z = 9$ and $t = 6$

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Solve the equation of x,y, z and t, if $2 [\begin{matrix} x & z y & t \end{matrix}] + 3 [\begin{matrix} 1 & - 1 0 & 2 \end{matrix}] = 3 [\begin{matrix} 3 & 5 4 & 6 \end{matrix}]$

Q. (i) If

A = [\begin{matrix} 1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1 \end{matrix}]

, find A⁻¹. Using A⁻¹, solve the system of linear equations
x − 2y = 10, 2x + y + 3z = 8, −2y + z = 7

(ii)

A = [\begin{matrix} 3 & - 4 & 2 \\ 2 & 3 & 5 \\ 1 & 0 & 1 \end{matrix}]

, find A⁻¹ and hence solve the following system of equations:
3x − 4y + 2z = −1, 2x + 3y + 5z = 7, x + z = 2

(iii)

A = [\begin{matrix} 1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1 \end{matrix}] and B = [\begin{matrix} 7 & 2 & - 6 \\ - 2 & 1 & - 3 \\ - 4 & 2 & 5 \end{matrix}]

, find AB. Hence, solve the system of equations:
x − 2y = 10, 2x + y + 3z = 8 and −2y + z = 7

(iv) If

A = [\begin{matrix} 1 & 2 & 0 \\ - 2 & - 1 & - 2 \\ 0 & - 1 & 1 \end{matrix}]

, find A⁻¹. Using A⁻¹, solve the system of linear equations
x − 2y = 10, 2x − y − z = 8, −2y + z = 7

(v) Given

A = [\begin{matrix} 2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5 \end{matrix}], B = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{matrix}]

, find BA and use this to solve the system of equations
y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17

(vi) If

A = (\begin{array}{rrr} 2 & 3 & 1 \\ 1 & 2 & 2 \\ - 3 & 1 & - 1 \end{array})

, find A^–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.
(vii) Use product

[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]

to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.

Q. Solve the equation for x , y , z and t if

Q. Solve the following equations:

x + 2 y + 2 z = 1

3 x - 2 y - 3 z = 0

x - 2 y + 2 z = 1

Q. Solve the following equations:
x-y=1
y+z=0
x-z=3

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Solve the equation for x,y,z and t if 2[xzyt]+3[1−102]=3[3546]

2[xzyt]+3[1−102]=3[3546]⇒[2x2z2y2t]+[3−306]=[9151218]⇒[2x+32z−32y2t+6]=[9151218]Comparing the corresponding elements of these two matrices, we get:2x+3=9⇒2x=6⇒x=32y=12⇒y=62z−3=15⇒2z=18⇒z=92t+6=18⇒2t=12t=6∴x=3,y=6,z=9 and t=6

Solve the equation for $x, y, z$ and $t$ if $2 [\begin{matrix} x & z y & t \end{matrix}] + 3 [\begin{matrix} 1 & - 1 0 & 2 \end{matrix}] = 3 [\begin{matrix} 3 & 5 4 & 6 \end{matrix}]$