Objective Questions And Answers On Mathematics

This set of Mathematics Objective Questions & Answers focuses on "Elementary Operation (Transformation) of a Matrix".

1. How many elementary operations are possible on Matrices?
a) 3
b) 2
c) 6
d) 5
View Answer

Answer: c
Explanation: There are a total of 6 elementary operations that are possible on matrices, three on rows and three on columns.

2. The following operation is applied on a matrix A=\(\begin{bmatrix}2&3\\6&4\end{bmatrix}\)
R₁→R₁+R₂
Which of the following will be the resulting new matrix?
a) \(\begin{bmatrix}8&7\\6&-4\end{bmatrix}\)
b) \(\begin{bmatrix}8&7\\6&4\end{bmatrix}\)
c) \(\begin{bmatrix}8&7\\6&5\end{bmatrix}\)
d) \(\begin{bmatrix}8&7\\6&2\end{bmatrix}\)
View Answer

Answer: b
Explanation: Given that, A=\(\begin{bmatrix}2&3\\6&4\end{bmatrix}\)
Applying the elementary operation, R₁→R₁+R₂ we get
B=\(\begin{bmatrix}2+6&3+4\\6&4\end{bmatrix}\)=\(\begin{bmatrix}8&7\\6&4\end{bmatrix}\).

3. Which of the following matrices will remain same if the elementary operation R₁→2R₁+3R₂ is applied on the matrix?
a) \(\begin{bmatrix}1&2&3\\3&4&1\end{bmatrix}\)
b) \(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)
c) \(\begin{bmatrix}0&1&0\\1&0&1\\0&1&0\end{bmatrix}\)
d) \(\begin{bmatrix}1&0\\1&2\\1&0\end{bmatrix}\)
View Answer

Answer: b
Explanation: Consider matrix A=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\), applying the elementary operation R₁→2R₁+3R₂.
\(\begin{bmatrix}2(0)+3(0)&2(0)+3(0)&2(0)+3(0)\\0&0&0\\0&0&0\end{bmatrix}\)=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\).
Therefore, the matrix A=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\), remains same after applying the elementary operation.

4. Which of the following is not a valid elementary operation?
a) R_i↔R_j
b) R_i→R_j+kR_i
c) R_i→kR_i
d) R_i→1+kR_i
View Answer

Answer: d
Explanation: The elementary operation R_i→1+kR_iis incorrect, the valid elementary operations on matrices are as follows.
i) Interchanging any two rows and columns
ii) The multiplication of the elements of any row or column by a non-zero number.
iii) The addition to the elements of any row or column, the corresponding elements of any other row and column multiplied by any non-zero number.

5. Which of the following elementary operations has been applied to the matrix A=\(\begin{bmatrix}8&5\\2&8\end{bmatrix}\) such that the new matrix is \(\begin{bmatrix}12&21\\2&8\end{bmatrix}\)?
a) R₁→R₁-2R₂
b) R₁→2R₁+R₂
c) R₁→R₂+R₁
d) R₁→R₁+2R₂
View Answer

Answer: d
Explanation: The given matrix is A=\(\begin{bmatrix}8&5\\2&8\end{bmatrix}\)
Applying the elementary operation R₁→R₁+2R₂, we get
\(\begin{bmatrix}8+2(2)&5+2(8)\\2&8\end{bmatrix}\)=\(\begin{bmatrix}12&21\\2&8\end{bmatrix}\).

6. The following elementary operations are applied to the matrix A=\(\begin{bmatrix}4&5&2\\6&7&1\\3&9&5\end{bmatrix}\)
R₁→2R₁+3R₂
R₂→3R₂-2R₃
Which among the following will be the new matrix?
a) \(\begin{bmatrix}24&31&7\\12&3&7\\3&9&5\end{bmatrix}\)
b) \(\begin{bmatrix}24&31&7\\12&3&-7\\3&9&5\end{bmatrix}\)
c) \(\begin{bmatrix}24&31&7\\6&7&1\\3&9&5\end{bmatrix}\)
d) \(\begin{bmatrix}4&5&2\\6&7&1\\3&9&5\end{bmatrix}\)
View Answer

Answer: b
Explanation: Given that, A=\(\begin{bmatrix}4&5&2\\6&7&1\\3&9&5\end{bmatrix}\)
Applying row operation, R₁→2R₁+3R₂
⇒\(\begin{bmatrix}2(4)+3(6)&2(5)+3(7)&2(2)+3(1)\\6&7&1\\3&9&5\end{bmatrix}\)=\(\begin{bmatrix}24&31&7\\6&7&1\\3&9&5\end{bmatrix}\)
Applying the row operation, R₂→3R₂-2R₃
⇒\(\begin{bmatrix}24&31&7\\3(6)-2(3)&3(7)-2(9)&3(1)-2(5)\\3&9&5\end{bmatrix}\)=\(\begin{bmatrix}24&31&7\\12&3&-7\\3&9&5\end{bmatrix}\)

7. The new matrix after applying the elementary operation R₂→2R₂+3R₁ on the matrix A=\(\begin{bmatrix}2&5&4\\5&2&6\\7&2&1\end{bmatrix}\) is _____________
a) \(\begin{bmatrix}2&5&4\\16&19&24\\7&2&1\end{bmatrix}\)
b) \(\begin{bmatrix}2&5&4\\19&19&24\\7&2&1\end{bmatrix}\)
c) \(\begin{bmatrix}2&-5&4\\16&19&24\\7&2&1\end{bmatrix}\)
d) \(\begin{bmatrix}1&5&4\\16&19&24\\7&2&1\end{bmatrix}\)
View Answer

Answer: a
Explanation: Consider A=\(\begin{bmatrix}2&5&4\\5&2&6\\7&2&1\end{bmatrix}\), after applying R₂→2R₂+3R₁
⇒\(\begin{bmatrix}2&5&4\\2(5)+3(2)&2(2)+3(5)&2(6)+3(4)\\7&2&1\end{bmatrix}\)=\(\begin{bmatrix}2&5&4\\16&19&24\\7&2&1\end{bmatrix}\).

8. Which among the following is the new matrix after applying the elementary operation C₁→4C₁ on the matrix A=\(\begin{bmatrix}5&8\\-1&2\\3&-4\end{bmatrix}\)?
a) \(\begin{bmatrix}5&8\\-1&2\\3&-4\end{bmatrix}\)
b) \(\begin{bmatrix}20&8\\-4&2\\12&-4\end{bmatrix}\)
c) \(\begin{bmatrix}20&8\\4&2\\12&-4\end{bmatrix}\)
d) \(\begin{bmatrix}20&8\\-4&2\\12&4\end{bmatrix}\)
View Answer

Answer: b
Explanation: Given matrix A=\(\begin{bmatrix}5&8\\-1&2\\3&-4\end{bmatrix}\) Applying the column operation, C₁→4C₁ we get
\(\begin{bmatrix}4(5)&8\\4(-1)&2\\4(3)&-4\end{bmatrix}\)=\(\begin{bmatrix}20&8\\-4&2\\12&-4\end{bmatrix}\)

9. The following column matrix operations are applied on a column matrix A=\(\begin{bmatrix}-7&2&6\\-2&3&-5\\2&1&3\end{bmatrix}\)
C₂→2C₁+C₂
C₃→3C₁+2C₃
Which among the following will be the new matrix?
a) \(\begin{bmatrix}-7&-12&6\\2&-1&-5\\2&-5&3\end{bmatrix}\)
b) \(\begin{bmatrix}-7&-12&6\\-2&-1&-5\\2&5&3\end{bmatrix}\)
c) \(\begin{bmatrix}-7&2&6\\-2&3&-5\\2&1&3\end{bmatrix}\)
d) \(\begin{bmatrix}-7&-12&-9\\-2&-1&-16\\2&5&12\end{bmatrix}\)
View Answer

Answer: d
Explanation: Given that, A=\(\begin{bmatrix}-7&2&6\\-2&3&-5\\2&1&3\end{bmatrix}\)
Applying the column operation, C₂→2C₁+C₂ we get
\(\begin{bmatrix}-7&2(-7)+2&6\\-2&2(-2)+3&-5\\2&2(2)+1&3\end{bmatrix}\)=\(\begin{bmatrix}-7&-12&6\\-2&-1&-5\\2&5&3\end{bmatrix}\)
Applying the column operation, C₃→3C₁+2C₃ we get
\(\begin{bmatrix}-7&-12&2(6)+3(-7)\\-2&-1&2(-5)+3(-2)\\2&5&2(3)+3(2)\end{bmatrix}\)=\(\begin{bmatrix}-7&-12&-9\\-2&-1&-16\\2&5&12\end{bmatrix}\)
Therefore, the resulting new matrix is \(\begin{bmatrix}-7&-12&-9\\-2&-1&-16\\2&5&12\end{bmatrix}\).

10. Which of the following column operation is incorrect for the matrix A=\(\begin{bmatrix}1&2&5\\6&3&8\end{bmatrix}\) ?
a) C₁→3C₁
b) C₂→C₁+C₂
c) C₂→2+2C₂
d) C₂→2C₁+2C₂-C₃
View Answer

Answer: c
Explanation: The column operation C₂→2+2C₂ is incorrect. A non-zero number cannot be directly added to any column or row in a matrix.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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