Using the property of determinants and without expanding- beginvmatrix 2 - 7 - 65 388875115 - 9 - 86lend vmatrix -0

Let A= 2 amp;7 amp;65 3 amp;8 amp;75 5 amp;9 amp;86 = 2 amp; 7 amp; 65 63 amp; 3 amp;8 amp;75 72 5 amp;9 ...

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Standard XII

Mathematics

Algebra of Complex Numbers

Using the pro...

Question

Using the property of determinants and without expanding.

$∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Solution

Let $A = ∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 - 63 3 & 8 & 75 - 72 5 & 9 & 86 - 81 \end{matrix} ∣ ∣ ∣ ∣$ (using $C_{3} \to C_{3} - 9 C_{2}$ )

$= ∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 2 3 & 8 & 3 5 & 9 & 5 \end{matrix} ∣ ∣ ∣ ∣ = 0$ [Since, the two columns $C_{3}$ and $C_{1}$ are identical]

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Using the property of determinants and without expanding, prove that

∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0

Q. Using the properties of determinant and without expanding , prove that:

∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0

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$∣ ∣ ∣ ∣ \begin{matrix} a - b & b - c & c - a b - c & c - a & a - b c - a & a - b & b - c \end{matrix} ∣ ∣ ∣ ∣ = 0$

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$∣ ∣ ∣ ∣ \begin{matrix} x & a & x + a y & b & y + b z & c & z + c \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Using the property of determinants and without expanding. ∣∣ ∣∣276538755986∣∣ ∣∣=0

Let A=∣∣ ∣∣276538755986∣∣ ∣∣=∣∣ ∣∣2765−633875−725986−81∣∣ ∣∣ (using C3→C3−9C2) =∣∣ ∣∣272383595∣∣ ∣∣=0 [Since, the two columns C3 and C1 are identical]

Using the property of determinants and without expanding.

$∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0$