Evaluate the determinants: $(i)$ $\left|\begin{array}{ccc}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$ (ii) $\left|\begin{array}{ccc}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$ (iii) $\left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$ (iv) $\left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$
(A) $(i)$ Let $A = \left|\begin{array}{ccc}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$. Expanding along the second row: $|A| = -0\left|\begin{array}{cc}-1 & -2 \\ -5 & 0\end{array}\right| + 0\left|\begin{array}{cc}3 & -2 \\ 3 & 0\end{array}\right| - (-1)\left|\begin{array}{cc}3 & -1 \\ 3 & -5\end{array}\right| = 1(-15 - (-3)) = -15 + 3 = -12$.
(ii) Let $A = \left|\begin{array}{ccc}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$. Expanding along the first row: $|A| = 3(1 - (-6)) - (-4)(1 - (-4)) + 5(3 - 2) = 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46$.
(iii) Let $A = \left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$. Expanding along the first row: $|A| = 0(0 - (-9)) - 1(0 - 6) + 2(-3 - 0) = 0 + 6 - 6 = 0$.
(iv) Let $A = \left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$. Expanding along the first column: $|A| = 2(0 - 5) - 0 + 3(1 - (-4)) = 2(-5) + 3(5) = -10 + 15 = 5$.
(ii) Let $A = \left|\begin{array}{ccc}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$. Expanding along the first row: $|A| = 3(1 - (-6)) - (-4)(1 - (-4)) + 5(3 - 2) = 3(7) + 4(5) + 5(1) = 21 + 20 + 5 = 46$.
(iii) Let $A = \left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$. Expanding along the first row: $|A| = 0(0 - (-9)) - 1(0 - 6) + 2(-3 - 0) = 0 + 6 - 6 = 0$.
(iv) Let $A = \left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$. Expanding along the first column: $|A| = 2(0 - 5) - 0 + 3(1 - (-4)) = 2(-5) + 3(5) = -10 + 15 = 5$.
Explore More
Similar Questions
Which mineral element plays a crucial role in nitrogen metabolism?
The work function of a metallic surface is $5.01 \ eV$. The photo-electrons are emitted when light of wavelength $2000 \ \mathring{A}$ falls on it. The potential difference applied to stop the fastest photo-electrons is ............. $volts$ $[h = 4.14 \times 10^{-15} \ eV \ s]$
The cathode of a photoelectric cell is changed such that the work function changes from $W_1$ to $W_2$ $(W_2 > W_1)$. If the current before and after the change are $I_1$ and $I_2$ respectively,with all other conditions remaining unchanged,then (assuming $h\nu > W_2$):
When a plane mirror is rotated through an angle $\theta$,the reflected ray turns through an angle $2\theta$. What happens to the size of the image?
Observe the following reaction: $2A + B \longrightarrow C$. The rate of formation of $C$ is $2.2 \times 10^{-3} \ mol \ L^{-1} \ min^{-1}$. What is the value of $-\frac{d[A]}{dt}$ (in $mol \ L^{-1} \ min^{-1}$)?
Vedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo