Higher Secondary Mathematics: 20.2

Saturday, March 30, 2024

20.2 - Solved Examples

In the previous section, we saw how to obtain the determinant of order 3. In this section, we will see some solved examples.

Solved example 20.3
Evaluate the determinant $\Delta \;=\; \left|\begin{array}{r} 1 &{ 2 } &{ 4 } \\ -1 &{ 3 } &{ 0 } \\ 4 &{ 1 } &{ 0 } \\ \end{array}\right|$
Solution:
We will expand along C₃ because, it has the maximum number of zeroes. We get:

Solved example 20.4
Evaluate

$\Delta \;=\; \left|\begin{array}{r} 0 &{ \sin \alpha } &{ -\cos \alpha } \\ -\sin \alpha &{ 0 } &{ \sin \beta } \\ \cos \alpha &{ -\sin \beta } &{ 0 } \\ \end{array}\right|$
Solution:
Expanding along R₁, we get:

Solved example 20.5
Find the values of x for which $\left|\begin{array}{r} 3 &{ x } \\ x &{ 1 } \\ \end{array}\right|~=~\left|\begin{array}{r} 3 &{ 2 } \\ 4 &{ 1 } \\ \end{array}\right|$
Solution:

Solved example 20.6
Find the values of x if
(i) $\left|\begin{array}{r} 2 &{ 4 } \\ 5 &{ 1 } \\ \end{array}\right|~=~\left|\begin{array}{r} 2x &{ 4 } \\ 6 &{ x } \\ \end{array}\right|$
(ii) $\left|\begin{array}{r} 2 &{ 3 } \\ 4 &{ 5 } \\ \end{array}\right|~=~\left|\begin{array}{r} x &{ 3 } \\ 2x &{ 5 } \\ \end{array}\right|$
Solution:
Part (i):

Part (ii):

Solved example 20.7
If $\left|\begin{array}{r} x &{ 2 } \\ 18 &{ x } \\ \end{array}\right|~=~\left|\begin{array}{r} 6 &{ 2 } \\ 18 &{ 6 } \\ \end{array}\right|$ , then x is equal to
(A) 6 (B) ∓ 6 (C) -6 (D) 0
Solution: