Determinants and their Properties
Matrix and Determinant

79037 If \(A\) is square matrix of order 3 , then consider the following statements.
I. If \(|\mathbf{A}|=\mathbf{0}\), then \(|\operatorname{Adj} \mathbf{A}|=\mathbf{0}\)
II. If \(|\mathbf{A}| \neq \mathbf{0}\), then \(\left|\mathbf{A}^{-1}\right|=|\mathbf{A}|^{-1}\)
Which of the above statements is/are true?

1 Both I and II
2 Neither I nor II
3 I only
4 II only
Matrix and Determinant

79038 If \(a_{1}, a_{2}, \ldots . . . a_{n}, \ldots .\). are in G.P. and \(a i,>0\) from each 1 , then the determinant
\(\Delta=\left|\begin{array}{ccc} \log a_n & \log a_{n+2} & \log a_{n+4} \\ \log a_{n+6} & \log a_{n+8} & \log a_{n+10} \\ \log a_{n+12} & \log a_{n+14} & \log a \end{array}\right|\) is equal to

1 0
2 1
3 2
4 n
Matrix and Determinant

79039 If \(A(\theta)=\left[\begin{array}{cc}i \sin \theta & \cos \theta \\ \cos \theta & \text { isin } \theta\end{array}\right]\) is a matrix where is \(i=\sqrt{-1}\), then which of the following is not true

1 \(\operatorname{det} \mathrm{A}(\pi+\theta)=\operatorname{det} \mathrm{A}(-\theta)\)
2 \(\operatorname{det} \mathrm{A}(-\theta)=\operatorname{det} \mathrm{A}(\theta)\)
3 \(\operatorname{det}[\mathrm{A}(\theta)]^{-1}=1\)
4 \(\operatorname{det} \mathrm{A}(-\theta)=-1\)
Matrix and Determinant

79040 The value of the determinant
\(\left|\begin{array}{ccc} a+b & a+2 b & a+3 b \\ a+2 b & a+3 b & a+4 b \\ a+4 b & a+5 b & a+6 b \end{array}\right| \text { is }\)

1 a
2 \(\mathrm{b}\)
3 0
4 \(a+b\)
Matrix and Determinant

79037 If \(A\) is square matrix of order 3 , then consider the following statements.
I. If \(|\mathbf{A}|=\mathbf{0}\), then \(|\operatorname{Adj} \mathbf{A}|=\mathbf{0}\)
II. If \(|\mathbf{A}| \neq \mathbf{0}\), then \(\left|\mathbf{A}^{-1}\right|=|\mathbf{A}|^{-1}\)
Which of the above statements is/are true?

1 Both I and II
2 Neither I nor II
3 I only
4 II only
Matrix and Determinant

79038 If \(a_{1}, a_{2}, \ldots . . . a_{n}, \ldots .\). are in G.P. and \(a i,>0\) from each 1 , then the determinant
\(\Delta=\left|\begin{array}{ccc} \log a_n & \log a_{n+2} & \log a_{n+4} \\ \log a_{n+6} & \log a_{n+8} & \log a_{n+10} \\ \log a_{n+12} & \log a_{n+14} & \log a \end{array}\right|\) is equal to

1 0
2 1
3 2
4 n
Matrix and Determinant

79039 If \(A(\theta)=\left[\begin{array}{cc}i \sin \theta & \cos \theta \\ \cos \theta & \text { isin } \theta\end{array}\right]\) is a matrix where is \(i=\sqrt{-1}\), then which of the following is not true

1 \(\operatorname{det} \mathrm{A}(\pi+\theta)=\operatorname{det} \mathrm{A}(-\theta)\)
2 \(\operatorname{det} \mathrm{A}(-\theta)=\operatorname{det} \mathrm{A}(\theta)\)
3 \(\operatorname{det}[\mathrm{A}(\theta)]^{-1}=1\)
4 \(\operatorname{det} \mathrm{A}(-\theta)=-1\)
Matrix and Determinant

79040 The value of the determinant
\(\left|\begin{array}{ccc} a+b & a+2 b & a+3 b \\ a+2 b & a+3 b & a+4 b \\ a+4 b & a+5 b & a+6 b \end{array}\right| \text { is }\)

1 a
2 \(\mathrm{b}\)
3 0
4 \(a+b\)
Matrix and Determinant

79037 If \(A\) is square matrix of order 3 , then consider the following statements.
I. If \(|\mathbf{A}|=\mathbf{0}\), then \(|\operatorname{Adj} \mathbf{A}|=\mathbf{0}\)
II. If \(|\mathbf{A}| \neq \mathbf{0}\), then \(\left|\mathbf{A}^{-1}\right|=|\mathbf{A}|^{-1}\)
Which of the above statements is/are true?

1 Both I and II
2 Neither I nor II
3 I only
4 II only
Matrix and Determinant

79038 If \(a_{1}, a_{2}, \ldots . . . a_{n}, \ldots .\). are in G.P. and \(a i,>0\) from each 1 , then the determinant
\(\Delta=\left|\begin{array}{ccc} \log a_n & \log a_{n+2} & \log a_{n+4} \\ \log a_{n+6} & \log a_{n+8} & \log a_{n+10} \\ \log a_{n+12} & \log a_{n+14} & \log a \end{array}\right|\) is equal to

1 0
2 1
3 2
4 n
Matrix and Determinant

79039 If \(A(\theta)=\left[\begin{array}{cc}i \sin \theta & \cos \theta \\ \cos \theta & \text { isin } \theta\end{array}\right]\) is a matrix where is \(i=\sqrt{-1}\), then which of the following is not true

1 \(\operatorname{det} \mathrm{A}(\pi+\theta)=\operatorname{det} \mathrm{A}(-\theta)\)
2 \(\operatorname{det} \mathrm{A}(-\theta)=\operatorname{det} \mathrm{A}(\theta)\)
3 \(\operatorname{det}[\mathrm{A}(\theta)]^{-1}=1\)
4 \(\operatorname{det} \mathrm{A}(-\theta)=-1\)
Matrix and Determinant

79040 The value of the determinant
\(\left|\begin{array}{ccc} a+b & a+2 b & a+3 b \\ a+2 b & a+3 b & a+4 b \\ a+4 b & a+5 b & a+6 b \end{array}\right| \text { is }\)

1 a
2 \(\mathrm{b}\)
3 0
4 \(a+b\)
Matrix and Determinant

79037 If \(A\) is square matrix of order 3 , then consider the following statements.
I. If \(|\mathbf{A}|=\mathbf{0}\), then \(|\operatorname{Adj} \mathbf{A}|=\mathbf{0}\)
II. If \(|\mathbf{A}| \neq \mathbf{0}\), then \(\left|\mathbf{A}^{-1}\right|=|\mathbf{A}|^{-1}\)
Which of the above statements is/are true?

1 Both I and II
2 Neither I nor II
3 I only
4 II only
Matrix and Determinant

79038 If \(a_{1}, a_{2}, \ldots . . . a_{n}, \ldots .\). are in G.P. and \(a i,>0\) from each 1 , then the determinant
\(\Delta=\left|\begin{array}{ccc} \log a_n & \log a_{n+2} & \log a_{n+4} \\ \log a_{n+6} & \log a_{n+8} & \log a_{n+10} \\ \log a_{n+12} & \log a_{n+14} & \log a \end{array}\right|\) is equal to

1 0
2 1
3 2
4 n
Matrix and Determinant

79039 If \(A(\theta)=\left[\begin{array}{cc}i \sin \theta & \cos \theta \\ \cos \theta & \text { isin } \theta\end{array}\right]\) is a matrix where is \(i=\sqrt{-1}\), then which of the following is not true

1 \(\operatorname{det} \mathrm{A}(\pi+\theta)=\operatorname{det} \mathrm{A}(-\theta)\)
2 \(\operatorname{det} \mathrm{A}(-\theta)=\operatorname{det} \mathrm{A}(\theta)\)
3 \(\operatorname{det}[\mathrm{A}(\theta)]^{-1}=1\)
4 \(\operatorname{det} \mathrm{A}(-\theta)=-1\)
Matrix and Determinant

79040 The value of the determinant
\(\left|\begin{array}{ccc} a+b & a+2 b & a+3 b \\ a+2 b & a+3 b & a+4 b \\ a+4 b & a+5 b & a+6 b \end{array}\right| \text { is }\)

1 a
2 \(\mathrm{b}\)
3 0
4 \(a+b\)