Exercise 2.3 (Solutions)

The solutions of the Exercise 2.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to determinant and inverse of the matrix.

Question. 1 Evaluate the determinant of the following matrices.

(i) $\left[\begin{array}{ccc}2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2\end{array}\right]$

(ii) $\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$

(iii) $\left[\begin{array}{ccc}i & 3 & -2 i \\ 1 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2+i & 1 & i \\ 0 & 2 & 1 \\ -3 i & 1 & 6\end{array}\right]$
Solution:Question 1

Question. 2 Evaluate the determinants of the following matrices using cofactor method.

(i) $\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 5 & 1 \\ 2 & 1 & 0\end{array}\right]$

(ii) $\left[\begin{array}{ccc}2 & 3 & -1 \\ -1 & 0 & 2 \\ 3 & 1 & 4\end{array}\right]$

(iii) $\left[\begin{array}{ccc}2 i & 6 & 1 \\ 1 & -i & 2 \\ 0 & 1 & 3 i\end{array}\right]$

(iv) $\left[\begin{array}{ccc}1-i & 2 & 1+i \\ 3 & 1 & 4 \\ 0 & 2 & 3\end{array}\right]$
Solution:Question 2

Question. 3 Determine which of the following matrices are singular and which are non-singular.

(i) $\left[\begin{array}{ccc}3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3\end{array}\right]$

(ii) $\left[\begin{array}{ccc}3 & -1 & 2 \\ 2 & 0 & 1 \\ -1 & 5 & 1\end{array}\right]$

(iii) $\left[\begin{array}{ccc}3 i & 1 & 2 \\ -4 & 1 & i \\ 2 & 0 & 1\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2 & -i & 1 \\ i & 3 & -2 \\ -2+i & i+3 & -3\end{array}\right]$
Solution:Question 3

Question. 4 Find the value of $\lambda$, so that the given matrices are singular.

(i) $\left[\begin{array}{lll}\lambda & 1 & 3 \\ 2 & 1 & 8 \\ 0 & 3 & 1\end{array}\right]$

(ii) $\left[\begin{array}{lll}\lambda & 2 & 0 \\ 2 & 1 & 3 \\ \lambda & 2 & 1\end{array}\right]$

(iii) $\left[\begin{array}{lll}\lambda & i & 1 \\ 2 & 1 & 3 \\ 3 & 1 & 2\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2+i & 1 & 6 \\ 2 & \lambda & 1 \\ 3 & 0 & 2\end{array}\right]$
Solution:Question 4

Question. 5 Find the multiplicative inverse of the following matrices if it exists by adjoint method.

(i) $\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1\end{array}\right]$

(ii) $\left[\begin{array}{ccc}3 & -4 & 2 \\ 2 & 3 & 5 \\ 1 & 0 & 1\end{array}\right]$

(iii) $\left[\begin{array}{ccc}i & 0 & 1 \\ 2 i & -1 & -i \\ 1 & 0 & 4 i\end{array}\right]$

(iv) $\left[\begin{array}{ccc}3 & -i & i \\ 2 & 1 & -3 i \\ 4 i & 2 & 6\end{array}\right]$
Solution:Question 5

Question. 6 If $A=\left[\begin{array}{ccc}2 & 1 & -3 \\ 0 & 1 & 0 \\ 2 & 1 & 6\end{array}\right]$ then find $A^{-1}$ and hence show that $A A^{-1}=A^{-1} A=I_{3}$.
Solution:Question 6

Question. 7 Verify that $(A B)^{-1}=B^{-1} A^{-1}$ in each of the following.

(i) $A=\left[\begin{array}{ll}2 & 1 \\ 8 & 6\end{array}\right]$ and $B=\left[\begin{array}{ll}3 & 2 \\ 0 & 2\end{array}\right]$

(ii) $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & -1 & 1 \\ 2 & 1 & -3\end{array}\right]$ and $B=\left[\begin{array}{ccc}3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2\end{array}\right]$

(iii) $A=\left[\begin{array}{ccc}2 & -i & 6 \\ 1 & 2 & i \\ -i & 1 & 6\end{array}\right]$ and $B=\left[\begin{array}{lll}3 & 1 & 2 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right]$

(iv) $A=\left[\begin{array}{ccc}1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1\end{array}\right]$ and $B=\left[\begin{array}{lll}2 & 3 & 4 \\ 1 & 0 & 2 \\ 0 & 1 & 3\end{array}\right]$
Solution:Question 7