Prove that- beginvmatrix a- 2-2 a - 2 a-1 - 1 2 a-1 - a-2 - 1113 - 3 - 1lend vmatrix - a -1- 3

We have to prove, = a^2+2a amp; 2a+1 amp; 1 2a+1 amp;a+2 amp;1 3 amp; 3 amp;1 =(a 1)^3 ∴ LHS= a^2+2a amp; 2a+1 amp; 1 2a+1 amp; ...

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Standard XII

Mathematics

Determinant

Prove that: b...

Question

Prove that:
$∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = (a - 1)^{3}$

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Solution

We have to prove,
$= ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = (a - 1)^{3} ∴ L H S = ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a - 2 a - 1 & 2 a + 1 - a - 2 & 0 2 a + 1 - 3 & a + 2 - 3 & 0 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ [∵ R_{1} \to R_{1} - R_{2} a n d R_{2} \to R_{2} - R_{3}] = ∣ ∣ ∣ ∣ ∣ \begin{matrix} (a - 1) (a + 1) & (a - 1) & 0 2 (a - 1) & (a - 1) & 0 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - 1)^{2} ∣ ∣ ∣ ∣ \begin{matrix} (a + 1) & 1 & 0 2 & 1 & 0 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣$
[taking (a - 1) common from $R_{1}$ and $R_{2}$ each]
$= (a - 1)^{2} [1 (a + 1) - 2] = (a - 1)^{3}$
=RHS
Hence proved.

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Similar questions

Q. Prove that:

∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = (a - 1)^{C}

then

C =

Q. Using properties of determinants, prove that

∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = (a - 1)^{3}

Using properties of determinants,prove that $∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = (a - 1)^{3}$ .

OR find matrix A such that $⎛ ⎜ ⎝ \begin{matrix} 2 & - 1 1 & 0 - 3 & 4 \end{matrix} ⎞ ⎟ ⎠ A = ⎛ ⎜ ⎝ \begin{matrix} - 1 & - 8 1 & - 2 9 & 22 \end{matrix} ⎞ ⎟ ⎠ .$

Q. Prove that

∣ ∣ ∣ ∣ \begin{matrix} a + b + 2 c & a & b c & b + c + 2 a & b c & a & c + a + 2 b \end{matrix} ∣ ∣ ∣ ∣ = 2 (a + b + c)^{3}

$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & x & x^{2} x^{2} & 1 & x x & x^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = (1 - x^{3})^{2}$

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Prove that: ∣∣ ∣∣a2+2a2a+112a+1a+21331∣∣ ∣∣=(a−1)3

Prove that:
$∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 2 a & 2 a + 1 & 1 2 a + 1 & a + 2 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ = (a - 1)^{3}$