4. If \(A=\begin{vmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{vmatrix}\), then show that |3A|=27|A|.
Class 12
Determinants
Exercise 4.1
Evaluate the determinants in Exercise 1 and 2.
1. \(\begin{vmatrix}2 & 4 \\ -5 & -1\end{vmatrix}\)
2. (i) \(\begin{vmatrix}cosθ & -sinθ \\ sinθ & cosθ\end{vmatrix}\)
(ii) \(\begin{vmatrix}x^2-x+1 & x-1 \\ x+1 & x+1\end{vmatrix}\)
3. If \(A=\begin{bmatrix}1 & 2 \\ 4 & 2\end{bmatrix}\), then show that |2A|=4|A|.
(i) \(\begin{vmatrix}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}\)
(ii) \(\begin{vmatrix}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}\)
(iii) \(\begin{vmatrix}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}\)
(iv) \(\begin{vmatrix}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{vmatrix}\)
6. If \(A=\begin{vmatrix}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{vmatrix}\), find |A|.
(i) \(\begin{vmatrix}2 & 4 \\5 & 1 \end{vmatrix}\)=\(\begin{vmatrix}2x & 4 \\6 & x \end{vmatrix}\)
(ii) \(\begin{vmatrix}2 & 3 \\4 & 5 \end{vmatrix}\)=\(\begin{vmatrix}x & 3 \\2x & 5 \end{vmatrix}\)