Given that: $A = ⎛ ⎜ ⎝ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎞ ⎟ ⎠$ and $B = ⎛ ⎜ ⎝ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎞ ⎟ ⎠$ , find AB.
Using this result, solve the following system of equation:
$x - y = 3, 2 x + 3 y + 4 z = 17$ and $y + 2 x = 7$

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Solution

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$

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

$A B = ⎡ ⎢ ⎣ \begin{matrix} 6 & 0 & 0 0 & 6 & 0 0 & 0 & 6 \end{matrix} ⎤ ⎥ ⎦$

$A B = D$

$x - y = 3$
$2 x + 3 y + 4 z = 17$
$y + 2 z = 7$

Converting it into matrix then we get ,
$⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ \times ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 3177 \end{matrix} ⎤ ⎥ ⎦$

$A X = C$
$(A B) X = B C$
$D X = B C$

$⎡ ⎢ ⎣ \begin{matrix} 6 x 6 y 6 z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & - 4 - 4 & 2 & - 4 2 & - 1 & 5 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 3177 \end{matrix} ⎤ ⎥ ⎦$

$⎡ ⎢ ⎣ \begin{matrix} 6 x 6 y 6 z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 12 - 6 24 \end{matrix} ⎤ ⎥ ⎦$

$6 x = 12, x = 2$
$6 y = - 6, y = - 1$
$6 z = 24, z = 4$

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Similar questions

Q. If

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

are two square matrices, find AB and hence solve the system of linear equations:
x − y = 3, 2x + 3y + 4z = 17, y + 2z = 7

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

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 2 & 3 & 4 0 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦

and

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

are two square matrices, find

A B

Q. Solve the system of equations:

y + 2 z = 7

x - y = 3

2 x + 3 y + 4 z = 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.

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Given that: A=⎛⎜⎝1−10234012⎞⎟⎠ and B=⎛⎜⎝22−4−42−42−15⎞⎟⎠, find AB.Using this result, solve the following system of equation:x−y=3,2x+3y+4z=17 and y+2x=7