Exercise 1.4 - Chapter 1 Applications of Matrices and Determinants 12th Maths Guide Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $1.4$
Choose the correct answer.
Question 1.

If $\mathrm{A}=\left(\begin{array}{lll}1 & 2 & 3\end{array}\right)$, then the rank of $\mathrm{AA}^{\mathrm{T}}$ is _______________
(a) 0
(b) 2
(c) 3
(d) 1
Answer:
(d) 1
Hint:
$
\begin{aligned}
& A=\left(\begin{array}{lll}
1 & 2 & 3
\end{array}\right) \quad A^T=\left(\begin{array}{l}
1 \\
2 \\
3
\end{array}\right) \\
& \begin{array}{l}
\mathrm{A} \mathrm{A}^{\mathrm{T}}=\mathrm{A}=\left(\begin{array}{lll}
1 & 2 & 3
\end{array}\right)\left(\begin{array}{l}
1 \\
2 \\
3
\end{array}\right)=(1+4+9)=(14) \\
\text { oo rank is 1 }
\end{array} \\
&
\end{aligned}
$
Question 2.
The rank of $\mathrm{m} \times \mathrm{n}$ matrix whose elements are unity is
(a) 0
(b) 1
(c) $\mathrm{m}$
(d) $n$
Answer:
(b) 1
Hint:
All the rows except the first row can be made zero

Question 3.

 is a transition probability matrix, then at equilibrium $\mathrm{A}$ is equal to
(a) $\frac{1}{4}$
(b) $\frac{1}{5}$
(c) $\frac{1}{6}$
(d) $\frac{1}{8}$

Answer:
(a) $\frac{1}{4}$
Hint:
$(A$B) $\left(\begin{array}{ll}0.4 & 0.6 \\ 0.2 & 0.8\end{array}\right)=\left(\begin{array}{ll}A & B\end{array}\right)$
$
\begin{aligned}
& (0.4 \mathrm{~A}+0.2 \mathrm{~B} 0.6 \mathrm{~A}+0.8 \mathrm{~B})=\left(\begin{array}{ll}
A \mathrm{~A} & \mathrm{~B}
\end{array}\right) \\
& 0.4 \mathrm{~A}+0.2(1-\mathrm{A})=\mathrm{A} \Rightarrow \mathrm{A}=\frac{0.2}{0.8}=\frac{1}{4} \\
&
\end{aligned}
$
Question 4.
If $\mathrm{A}=\left(\begin{array}{ll}2 & 0 \\ 0 & 8\end{array}\right)$ then $\rho(A)$ is
(a) 0
(b) 1
(c) 2
(d) $\mathrm{n}$
Answer:
(c) 2
Hint:
$
A=\left(\begin{array}{ll}
2 & 0 \\
0 & 8
\end{array}\right) \quad\left|\begin{array}{ll}
2 & 0 \\
0 & 8
\end{array}\right|=16 \neq 0 \quad \text { So, } \rho(A)=2
$
Question 5.
The rank of the matrix $\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9\end{array}\right)$ is
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(d) 3
Hint:

So rank is 3

Question 6.
The rank of the unit matrix of order $\mathrm{n}$ is
(a) $\mathrm{n}-1$
(b) $\mathrm{n}$
(c) $\mathrm{n}+1$
(d) $\mathrm{n}^2$
Answer:
(b) $\mathrm{n}$
Hint:
Unit matrix of order $\mathrm{n}$ is in echelon form with $\mathrm{n}$ non-zero rows
Question 7.
If $\rho(A)=r$ then which of the following is correct?
(a) all the minors of order $r$ which does not vanish
(b) A has at least one minor of order $r$ which does not vanish
(c) A has at least one $(r+1)$ order minor which vanishes
(d) all $(r+1)$ and higher-order minors should not vanish
Answer:
(b) A has at least one minor of order $r$ which does not vanish
Question 8.
If $\mathrm{A}=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ then the rank of $\mathrm{AA}^{\mathrm{T}}$ is
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(b) 1
Hint:
$
\begin{aligned}
& A=\left(\begin{array}{l}
1 \\
2 \\
3
\end{array}\right) \Rightarrow A^{\mathrm{T}}=\left(\begin{array}{lll}
1 & 2 & 3
\end{array}\right) \\
& \mathrm{AA}^{\mathrm{T}}=\left(\begin{array}{l}
1 \\
2 \\
3
\end{array}\right) \quad\left(\begin{array}{lll}
1 & 2 & 3
\end{array}\right)=\left(\begin{array}{lll}
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & 6 & 9
\end{array}\right) \\
& \sim\left(\begin{array}{lll}
1 & 2 & 3 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right) \quad \begin{array}{l}
\mathrm{R}_2 \rightarrow \mathrm{R}_2-2 \mathrm{R}_1 \\
\mathrm{R}_3 \rightarrow \mathrm{R}_3-3 \mathrm{R}_1
\end{array} \\
&
\end{aligned}
$

So rank is 1
Question 9.
If the rank of the matrix $\left(\begin{array}{ccc}\lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda\end{array}\right)$ is 2 . Then $\lambda$ is
(a) 1
(b) 2
(c) 3
(d) only real number
Answer:
(a) 1
Hint:

Since rank is 2 , the third order minor should vanish.
$
\begin{aligned}
& \lambda^3-1=0 \\
& \Rightarrow \lambda=1
\end{aligned}
$

Question 10.

The rank of the diagonal matrixis
(a) 0
(b) 2
(c) 3
(d) 5
Answer:
(c) 3
Hint:
There are only three non-zero rows as the matrix is in echelon form.

Question 11.

If is a transition probability matrix, then the value of $\mathrm{x}$ is
(a) $0.2$
(b) $0.3$
(c) $0.4$
(d) $0.7$
Answer:
(c) $0.4$
Hint:
$
\mathrm{x}=1-0.6=0.4
$
Question 12.
Which of the following is not an elementary transformation?
(a) $\mathrm{R}_{\mathrm{i}} \leftrightarrow \mathrm{R}_{\mathrm{j}}$
(b) $\mathrm{R}_{\mathrm{i}} \rightarrow 2 \mathrm{R}_{\mathrm{i}}+2 \mathrm{C}_{\mathrm{j}}$
(c) $\mathrm{R}_{\mathrm{i}} \rightarrow 2 \mathrm{R}_{\mathrm{i}}-4 \mathrm{R}_{\mathrm{j}}$
(d) $\mathrm{C}_{\mathrm{i}} \rightarrow \mathrm{C}_{\mathrm{i}}+5 \mathrm{C}_{\mathrm{j}}$
Answer:
(b) $\mathrm{R}_{\mathrm{i}} \rightarrow 2 \mathrm{R}_{\mathrm{i}}+2 \mathrm{C}_{\mathrm{j}}$
Hint:
Since rows and columns cannot be taken together.
Question 13.
If $\rho(A)=\rho(A, B)$, then the system is
(a) Consistent and has infinitely many solutions
(b) Consistent and has unique solutions
(c) consistent
(d) inconsistent
Answer:
(c) consistent
Question 14.
If $\rho(A)=\rho(A, B)=$ the number of unknowns, then the system is
(a) Consistent and has infinitely many solutions
(b) Consistent and has unique solutions
(c) inconsistent

(d) consistent
Answer:
(i) Consistent and has unique solutions
Question 15.
If $\rho(A) \neq \rho(A, B)$, then the system is
(a) Consistent and has infinitely many solutions
(b) Consistent and has unique solutions
(c) inconsistent
(d) consistent
Answer:
(c) inconsistent
Question 16.
In a transition probability matrix, all the entries are greater than or equal to
(a) 2
(b) 1
(c) 0
(d) 3
Answer:
(c) 0
Question 17.
If the number of variables in a non- homogeneous system $A X=B$ is $n$, then the system possesses a unique solution only when
(a) $\rho(\mathrm{A})=\rho(\mathrm{A}, \mathrm{B})>\mathrm{n}$
(b) $\rho(\mathrm{A})=\rho(\mathrm{A}, \mathrm{B})=\mathrm{n}$
(c) $\rho(\mathrm{A})=\rho(\mathrm{A}, \mathrm{B})<\mathrm{n}$
(d) none of these
Answer:
(b) $\rho(\mathrm{A})=\rho(\mathrm{A}, \mathrm{B})=\mathrm{n}$
Question 18.
The system of equations $4 x+6 y=5,6 x+9 y=7$ has
(a) a unique solution

(c) infinitely many solutions
(d) none of these
Answer:

(b) no solution

Question 19.
For the system of equations $x+2 y+3 z=1,2 x+y+3 z=2,5 x+5 y+9 z=4$
(a) there is only one solution
(b) there exists infinitely many solutions
(c) there is no solution
(d) none of these
Answer:
(a) there is only one solution
Hint:

By Cramer's rule, there is only one solution
Question 20 .
If $|\mathrm{A}| \neq 0$, then $\mathrm{A}$ is
(a) non- singular matrix
(b) singular matrix
(c) zero matrix
(d) none of these
Answer:
(a) non-singular matrix
Question 21.
The system of linear equations $\mathrm{x}+\mathrm{y}+\mathrm{z}=2,2 \mathrm{x}+\mathrm{y}-\mathrm{z}=3,3 \mathrm{x}+2 \mathrm{y}+\mathrm{k}=4$ has unique solution, if $\mathrm{k}$ is not equal to
(a) 4
(b) 0
(c) $-4$
(d) 1
Answer:
(b) 0
Hint:

Question 22.
Cramer's rule is applicable only to get an unique solution when
(a) $\Delta_{\mathrm{z}} \neq 0$
(b) $\Delta_{\mathrm{x}} \neq 0$
(c) $\Delta \neq 0$
(d) $\Delta_{\mathrm{y}} \neq 0$
Answer:
(c) $\Delta \neq 0$
Question 23.
If $\frac{a_1}{x}+\frac{b_1}{y}=c_1, \frac{a_2}{x}+\frac{b_2}{y}=c_2, \Delta_1=\left|\begin{array}{ll}a_1 & b_1 \\ a_2 & b_2\end{array}\right| ; \Delta_2=\left|\begin{array}{ll}b_1 & c_1 \\ b_2 & c_2\end{array}\right| ; \Delta_3=\left|\begin{array}{ll}c_1 & a_1 \\ c_2 & a_2\end{array}\right|$ then $(x, y)$ is
(a) $\left(\frac{\Delta_2}{\Delta_1}, \frac{\Delta_3}{\Delta_1}\right)$
(b) $\left(\frac{\Delta_3}{\Delta_1}, \frac{\Delta_2}{\Delta_1}\right)$
(c) $\left(\frac{\Delta_1}{\Delta_2}, \frac{\Delta_1}{\Delta_3}\right)$
(d) $\left(\frac{-\Delta_1}{\Delta_2}, \frac{-\Delta_1}{\Delta_3}\right)$
Answer:
(d) $\left(\frac{-\Delta_1}{\Delta_2}, \frac{-\Delta_1}{\Delta_3}\right)$
Hint:
$
\begin{aligned}
& a_1\left(\frac{1}{x}\right)+b_1\left(\frac{1}{y}\right)=c_1 ; a_2\left(\frac{1}{x}\right)+b_2\left(\frac{1}{y}\right)=c_2 \\
& \frac{1}{x}=\frac{\left|\begin{array}{ll}
c_1 & b_1 \\
c_2 & b_2
\end{array}\right|}{\left|\begin{array}{ll}
a_1 & b_1 \\
a_2 & b_2
\end{array}\right|}=\frac{-\Delta_2}{\Delta_1} \\
& x=\frac{\Delta_1}{-\Delta_2}, \text { similarly } y=\frac{\Delta_1}{-\Delta_3}
\end{aligned}
$

Question 24.
$\left|\mathrm{A}_{\mathrm{n} \times \mathrm{n}}\right|=3|\operatorname{adj} \mathrm{A}|=243$ then the value $\mathrm{n}$ is
(a) 4
(b) 5
(c) 6
(d) 1
Answer:
(b) 5
Hint:
$|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{\mathrm{n}-1}, \mathrm{n}$ is order of matrix
$
\begin{aligned}
& 243=3^{\mathrm{n}-1} \\
& 3^4=3^{\mathrm{n}-1} \\
& \mathrm{n}=5
\end{aligned}
$
Question 25.
Rank of a null matrix is
(a) 0
(b) $-1$
(c) $\infty$
(d) 1
Answer:
(a) 0