If A - - 1 1 1- 1 2 3- 1 3 4 - B - - 7- 16- 22 - X - - x- y- z - and AX - B then z is equal to

The correct option is B 3 AX=B #160; 1 amp;1 #160; amp;1 #160; 1 amp; 2 amp; 3 1 amp;3 #160; amp;4 #160; ...

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Byju's Answer

Standard XII

Mathematics

Consistency of Linear System of Equations

If A = [ 1 ...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & 2 & 3 1 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 71622 \end{matrix} ⎤ ⎥ ⎦, X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦$ and $A X = B$ then $z$ is equal to

1

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Solution

The correct option is B $3$
$A X = B$
$∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 1 & 2 & 3 1 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣$ $∣ ∣ ∣ ∣ \begin{matrix} x y z \end{matrix} ∣ ∣ ∣ ∣$ $= ∣ ∣ ∣ ∣ \begin{matrix} x + y + z x + 2 y + 3 z x + 3 y + 4 z \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 71622 \end{matrix} ∣ ∣ ∣ ∣$
$\Rightarrow x + y + z = 7 . . . . . (1)$
$x + 2 y + 3 z = 16 . . . . . . (2)$
$x + 3 y + 4 z = 22 . . . . . . (3)$
$(2) - (1)$
$y + 2 z = 9 . . . . . . (4)$
$(3) - (2); y + z = 6 . . . . . . (5)$
$(4) - (5) z = 3$

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Similar questions

(i) If a(y+z) = b(z+x) = c(x+y) and out of a, b, c no two of them are equal then show that,

\frac{y - z}{a (b - c)} = \frac{z - x}{b (c - a)} = \frac{x - y}{c (a - b)}

(ii) If

\frac{x}{3 x - y - z} = \frac{y}{3 y - z - x} = \frac{z}{3 z - x - y}

and

x + y + z \neq 0

then show that the value of each ratio is equal to 1.

(iii) If

\frac{a x + b y}{x + y} = \frac{b x + a z}{x + z} = \frac{a y + b z}{y + z}

and

x + y + z \neq 0

then show that

\frac{a + b}{2}

.

(iv) If

\frac{y + z}{a} = \frac{z + x}{b} = \frac{x + y}{c}

then show that

\frac{x}{b + c - a} = \frac{y}{c + a - b} = \frac{z}{a + b - c}

.

(v) If

\frac{3 x - 5 y}{5 z + 3 y} = \frac{x + 5 z}{y - 5 x} = \frac{y - z}{x - z}

then show that every ratio =

\frac{x}{y}

Q. If the lines

x = 1 + a, y = - 3 - λ a, z = 1 + λ a

and

x = \frac{b}{2}, y = 1 + b, z = 2 - b

are coplanar, then

λ

is equal to

Q. If

s i n^{- 1} x + s i n^{- 1} y + s i n^{- 1} z = \frac{3 π}{2}

and

f (1) = 2, f (a + b) = f (a) f (b)

for all

a, b, ϵ R

, then

x^{f (1)} + y^{f (2)} + z^{f (3)} - \frac{x + y + z}{x^{f (1)} + y^{f (2)} + z^{f (3)}}

is equal to :

Q. If

\frac{x}{y + z} = a; \frac{y}{z + x} = b; \frac{z}{x + y} = c;

then

\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c}

is equal to

Q. If

x = 2, y = 1

and

z = - 3

, then

x^{3} + y^{3} + z^{3} - 3 x y z

is equal to

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If A=⎡⎢⎣111123134⎤⎥⎦,B=⎡⎢⎣71622⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦ and AX=B then z is equal to

The correct option is B 3AX=B∣∣ ∣∣111123134∣∣ ∣∣ ∣∣ ∣∣xyz∣∣ ∣∣ =∣∣ ∣∣x+y+zx+2y+3zx+3y+4z∣∣ ∣∣=∣∣ ∣∣71622∣∣ ∣∣⇒x+y+z=7.....(1)x+2y+3z=16......(2)x+3y+4z=22......(3)(2)−(1)y+2z=9......(4)(3)−(2);y+z=6......(5)(4)−(5)z=3

If $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & 2 & 3 1 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 71622 \end{matrix} ⎤ ⎥ ⎦, X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦$ and $A X = B$ then $z$ is equal to