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Matrices and DeterminantsQuestion and Answers: Page 3

Question Number 175110 Answers: 2 Comments: 3

Prove that determinant ((1,1,1),(a,b,c),(a^2 ,b^2 ,c^2 ))= (a−b)(b−c)(c−a)

$Prove that | \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix} | = (a - b) (b - c) (c - a)$

Question Number 175050 Answers: 0 Comments: 0

Question Number 174765 Answers: 0 Comments: 0

Question Number 172350 Answers: 0 Comments: 0

Question Number 172005 Answers: 2 Comments: 0

find x: (log_(10) x)^2 −log_(10) x=0

$f i n d x : {({l o g}_{10} x)}^{2} - {l o g}_{10} x = 0$

Question Number 171441 Answers: 0 Comments: 1

I_n = −((2n)/(2n + 1)) I_(n−1) I_0 = 1 Show that I_n = (((−4)^n (n!)^2 )/((2n+1)!))

$I_{n} = - \frac{2 n}{2 n + 1} I_{n - 1}$ $I_{0} = 1$ $S h o w t h a t I_{n} = \frac{{(- 4)}^{n} {(n!)}^{2}}{(2 n + 1)!}$

Question Number 171064 Answers: 2 Comments: 0

When A^(−1) = [(3,1),(8,4) ] find the A=? ,∣A^(−1) ∣∙A=?

$W h e n A^{- 1} = [\begin{matrix} 3 & 1 \\ 8 & 4 \end{matrix}]$ $f i n d t h e A = ?, ∣ A^{- 1} ∣ \cdot A = ?$

Question Number 170022 Answers: 0 Comments: 0

2. [((1 2)),((2 −1)) ] Soln: [((1 2)),((2 −1)) ]= [((1 0)),((0 1)) ].A ⇒ [((1 2)),((0 −5)) ]= [(( 1 0)),((−2 1)) ].A [R_2 →R_2 −2R_1 ] ⇒ [((1 2)),((0 1)) ]= [(( 1 0)),(((2/5) −(1/5))) ].A [R_2 →(−(1/5))R_2 ] ⇒ [((1 0)),((0 1)) ]= [(((1/5) (2/5))),(((2/5) −(1/5))) ].A [R_1 →R_1 −2R_2 ] ∴A^(−1) =(1/5) [((1 2)),((2 −1)) ]

$2 . [\begin{matrix} 1 2 \\ 2 - 1 \end{matrix}]$ $S o l n : [\begin{matrix} 1 2 \\ 2 - 1 \end{matrix}] = [\begin{matrix} 1 0 \\ 0 1 \end{matrix}] . A$ $\Rightarrow [\begin{matrix} 1 2 \\ 0 - 5 \end{matrix}] = [\begin{matrix} 1 0 \\ - 2 1 \end{matrix}] . A [R_{2} \to R_{2} - 2 R_{1}]$ $\Rightarrow [\begin{matrix} 1 2 \\ 0 1 \end{matrix}] = [\begin{matrix} 1 0 \\ \frac{2}{5} - \frac{1}{5} \end{matrix}] . A [R_{2} \to (- \frac{1}{5}) R_{2}]$ $\Rightarrow [\begin{matrix} 1 0 \\ 0 1 \end{matrix}] = [\begin{matrix} \frac{1}{5} \frac{2}{5} \\ \frac{2}{5} - \frac{1}{5} \end{matrix}] . A [R_{1} \to R_{1} - 2 R_{2}]$ $∴ A^{- 1} = \frac{1}{5} [\begin{matrix} 1 2 \\ 2 - 1 \end{matrix}]$

Question Number 168339 Answers: 0 Comments: 0

A point in rectangular coordinates (x,y,z) can be represented in spherical coordinates (r,θ,ϕ) by: x = r sin θ sin ϕ, y = sin θ sin ϕ, z = sin ϕ, 0 ≤ θ ≤ 2π , 0 ≤ ϕ ≤ π (a) Calculate the Jacobian of the transformation ((∂(x,y,z))/(∂(r,θ,ϕ))) (b) Calculate the volume of the region delimited by the sphere: S = {x,y,z ∈R^3 , x^2 +y^2 +z^2 ≤ R^2 , R>0}

$A point in rectangular coordinates$ $(x, y, z) can be represented in spherical$ $coordinates (r, θ, φ) by :$ $x = r \sin θ \sin φ, y = \sin θ \sin φ,$ $z = \sin φ, 0 ⩽ θ ⩽ 2 π, 0 ⩽ φ ⩽ π$ $(a) Calculate the Jacobian of the$ $transformation \frac{\partial (x, y, z)}{\partial (r, θ, φ)}$ $(b) Calculate the volume of the region$ $delimited by the sphere :$ $S = {x, y, z \in R^{3}, x^{2} + y^{2} + z^{2} ⩽ R^{2}, R > 0}$