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Chapter:

matrices

1. Which of following is inverse of matrix A=\(\begin{bmatrix}8&1\\1&2\end{bmatrix}\)?

\(\begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\\frac{1}{15}&\frac{8}{15}\end{bmatrix}\) \(\begin{bmatrix}\frac{1}{15}&-\frac{1}{15}\\-\frac{1}{15}&\frac{1}{15}\end{bmatrix}\) \(\begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\-\frac{1}{15}&\frac{8}{15}\end{bmatrix}\) \(\begin{bmatrix}\frac{2}{15}&\frac{1}{15}\\\frac{1}{15}&\frac{4}{15}\end{bmatrix}\)

\(\begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\-\frac{1}{15}&\frac{8}{15}\end{bmatrix}\)

Consider matrix A=\(\begin{bmatrix}8&1\\1&2\end{bmatrix}\)
Using elementary row operation, we write A=IA
Applying R₂→8R₂-R₁ and R₂→R₂/15, we get
\(\begin{bmatrix}8&1\\0&1\end{bmatrix}\)=\(\begin{bmatrix}1&0\\-\frac{1}{15}&\frac{8}{15}\end{bmatrix}\)A
Applying R₁→R₁-R₂ and R₁→R₁/8, we get
\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)=\(\begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\-\frac{1}{15}&\frac{8}{15}\end{bmatrix}\)A
A^-1=\(\begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\-\frac{1}{15}&\frac{8}{15}\end{bmatrix}\).

Hint Answer

2. Consider matrix A=\(\begin{bmatrix}2&4&5\\1&6&4\\2&8&9\end{bmatrix}\). Find element a₃₂.

5 6 4 8

8

a₃₂ is element represented in form a_ij i is row number and j is column number. Therefore, element a₃₂ is element in third row (i=3)and second column (j=2) which is 8.

Hint Answer

3. Which of following is a diagonal matrix.

A=\(\begin{bmatrix}0&2&1\\2&0&1\\2&1&0\end{bmatrix}\) A=\(\begin{bmatrix}5&1&0\\0&5&0\\0&0&5\end{bmatrix}\) A=\(\begin{bmatrix}4&0&0\\0&5&0\\0&0&9\end{bmatrix}\) A=\(\begin{bmatrix}2&2&2\\3&3&3\\4&4&4\end{bmatrix}\)

A=\(\begin{bmatrix}4&0&0\\0&5&0\\0&0&9\end{bmatrix}\)

The matrix is said to be a diagonal matrix if elements along diagonal of matrix are non ? zero.
i.e. a_ij=0 for i?j and a_ij?0 for i=j.
Therefore, matrix A=\(\begin{bmatrix}4&0&0\\0&5&0\\0&0&9\end{bmatrix}\) is a diagonal matrix.

Hint Answer

4. The matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is ______

scalar matrix identity matrix symmetric matrix skew-symmetric matrix

skew-symmetric matrix

The given matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is skew symmetric.
?A’=\(\begin{bmatrix}0&-1&1\\1&0&-1\\-1&1&0\end{bmatrix}\)=A
?A=-A’. Hence, it is a skew-symmetric matrix.

Hint Answer

5. Matrix addition and matrix multiplication both are commutative.

True False

False

The given statement is false. Matrix addition is commutative i.e. A+B=B+A. But matrix multiplication is not commutative i.e.AB?BA.

Hint Answer

6. Which of following matrices will remain same if elementary operation R₁→2R₁+3R₂ is applied on matrix?

\(\begin{bmatrix}1&2&3\\3&4&1\end{bmatrix}\) \(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\) \(\begin{bmatrix}0&1&0\\1&0&1\\0&1&0\end{bmatrix}\) \(\begin{bmatrix}1&0\\1&2\\1&0\end{bmatrix}\)

\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\)

Consider matrix A=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\), applying 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, matrix A=\(\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\), remains same after applying elementary operation.

Hint Answer

7. The matrix A=\(\begin{bmatrix}4\\12\\36\end{bmatrix}\) is _________

row matrix scalar matrix horizontal matrix column matrix

column matrix

The given matrix A = \(\begin{bmatrix}4\\12\\36\end{bmatrix}\) is of order 3�1. The matrix has only one column (n=1). Hence, it is a column matrix.

Hint Answer

8. The new matrix after applying elementary operation R₂→2R₂+3R₁ on matrix A=\(\begin{bmatrix}2&5&4\\5&2&6\\7&2&1\end{bmatrix}\) is _________

\(\begin{bmatrix}2&5&4\\16&19&24\\7&2&1\end{bmatrix}\) \(\begin{bmatrix}2&5&4\\19&19&24\\7&2&1\end{bmatrix}\) \(\begin{bmatrix}2&-5&4\\16&19&24\\7&2&1\end{bmatrix}\) \(\begin{bmatrix}1&5&4\\16&19&24\\7&2&1\end{bmatrix}\)

\(\begin{bmatrix}2&5&4\\16&19&24\\7&2&1\end{bmatrix}\)

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

Hint Answer

9. Which of following column operation is incorrect for matrix A=\(\begin{bmatrix}1&2&5\\6&3&8\end{bmatrix}\) ?

C₁→3C₁ C₂→C₁+C₂ C₂→2+2C₂ C₂→2C₁+2C₂-C₃

C₂→2+2C₂

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

Hint Answer

10. Find transpose of matrix A=\(\begin{bmatrix}-1&2&\sqrt{3}\\-4&5&\sqrt{6}\\-7&8&-9\end{bmatrix}\)

\(\begin{bmatrix}1&-2&-\sqrt{3}\\4&-5&-\sqrt{6}\\7&-8&9\end{bmatrix}\) \(\begin{bmatrix}-1&-4&-7\\2&5&8\\\sqrt{3}&\sqrt{6}&-9\end{bmatrix}\) \(\begin{bmatrix}1&4&7\\-2&-5&-8\\-\sqrt{3}&-\sqrt{6}&9\end{bmatrix}\) \(\begin{bmatrix}1&4&7\\-2&5&2\\1&8&9\end{bmatrix}\)

\(\begin{bmatrix}-1&-4&-7\\2&5&8\\\sqrt{3}&\sqrt{6}&-9\end{bmatrix}\)

To find transpose of matrix of given matrix, interchange rows with columns and columns with rows.
Hence, we get A’=\(\begin{bmatrix}-1&-4&-7\\2&5&8\\\sqrt{3}&\sqrt{6}&-9\end{bmatrix}\)

Hint Answer

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Topics

This Chapter matrices consists of the following topics

Matrices

Which of following is inverse of matrix A=\(\begin{bmatrix}8&1\\1&2\end{bmatrix}\)?

Consider matrix A=\(\begin{bmatrix}2&4&5\\1&6&4\\2&8&9\end{bmatrix}\). Find element a₃₂.

Which of following is a diagonal matrix.

The matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is ______

Matrix addition and matrix multiplication both are commutative.

Which of following matrices will remain same if elementary operation R₁→2R₁+3R₂ is applied on matrix?

The matrix A=\(\begin{bmatrix}4\\12\\36\end{bmatrix}\) is _________

The new matrix after applying elementary operation R₂→2R₂+3R₁ on matrix A=\(\begin{bmatrix}2&5&4\\5&2&6\\7&2&1\end{bmatrix}\) is _________

Which of following column operation is incorrect for matrix A=\(\begin{bmatrix}1&2&5\\6&3&8\end{bmatrix}\) ?

Find transpose of matrix A=\(\begin{bmatrix}-1&2&\sqrt{3}\\-4&5&\sqrt{6}\\-7&8&-9\end{bmatrix}\)

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