79103
If \(A\) is a square matrix of order 3 and \(\alpha\) is a real number, then determinant \([\alpha \mathrm{A}]\) is equal to
1 \(\alpha^{2}|\mathrm{~A}|\)
2 \(\alpha|\mathrm{A}|\)
3 \(\alpha^{3} \mid \mathrm{A}\)
4 None of these
Explanation:
(C) : Given, \(A\) is square matrix of order 3 and \(\alpha\) is a real number. We know that, if \(A\) is a square matrix of order \(n\) and its determinates if \(|\mathrm{A}|\) And for any scaler \(\mathrm{k},|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|\) Here, \(\mathrm{n}=3\) and \(\mathrm{k}=\alpha\) So, \(|\alpha \mathrm{A}|=\alpha^{3}|\mathrm{~A}|\)
J&K CET-2011
Matrix and Determinant
79104
The value of the determinant \(\left|\begin{array}{lll}4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3}\end{array}\right|=\) is
79106
If \(M\) and \(N\) are square matrices of order 3 where \(\operatorname{det}(M)=2\) and \(\operatorname{det}(N)=3\), then \(\operatorname{det}(3 \mathrm{MN})\) is
1 27
2 81
3 162
4 324
5 121
Explanation:
(C) : We know that, \(|\mathrm{MN}|=|\mathrm{M}||\mathrm{N}|=2 \times 3=6\) Also, for a square matrix of order 3 \(|\mathrm{KA}|=\mathrm{K}^{3}|\mathrm{~A}|\) because each element of the matrix \(\mathrm{A}\) is multiplied by \(\mathrm{k}\) and hence, \((3 \mathrm{MN})=(3)^{3}|\mathrm{MN}|\) \(=27 \times 6\) \(|3 \mathrm{MN}|=162\)
79103
If \(A\) is a square matrix of order 3 and \(\alpha\) is a real number, then determinant \([\alpha \mathrm{A}]\) is equal to
1 \(\alpha^{2}|\mathrm{~A}|\)
2 \(\alpha|\mathrm{A}|\)
3 \(\alpha^{3} \mid \mathrm{A}\)
4 None of these
Explanation:
(C) : Given, \(A\) is square matrix of order 3 and \(\alpha\) is a real number. We know that, if \(A\) is a square matrix of order \(n\) and its determinates if \(|\mathrm{A}|\) And for any scaler \(\mathrm{k},|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|\) Here, \(\mathrm{n}=3\) and \(\mathrm{k}=\alpha\) So, \(|\alpha \mathrm{A}|=\alpha^{3}|\mathrm{~A}|\)
J&K CET-2011
Matrix and Determinant
79104
The value of the determinant \(\left|\begin{array}{lll}4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3}\end{array}\right|=\) is
79106
If \(M\) and \(N\) are square matrices of order 3 where \(\operatorname{det}(M)=2\) and \(\operatorname{det}(N)=3\), then \(\operatorname{det}(3 \mathrm{MN})\) is
1 27
2 81
3 162
4 324
5 121
Explanation:
(C) : We know that, \(|\mathrm{MN}|=|\mathrm{M}||\mathrm{N}|=2 \times 3=6\) Also, for a square matrix of order 3 \(|\mathrm{KA}|=\mathrm{K}^{3}|\mathrm{~A}|\) because each element of the matrix \(\mathrm{A}\) is multiplied by \(\mathrm{k}\) and hence, \((3 \mathrm{MN})=(3)^{3}|\mathrm{MN}|\) \(=27 \times 6\) \(|3 \mathrm{MN}|=162\)
79103
If \(A\) is a square matrix of order 3 and \(\alpha\) is a real number, then determinant \([\alpha \mathrm{A}]\) is equal to
1 \(\alpha^{2}|\mathrm{~A}|\)
2 \(\alpha|\mathrm{A}|\)
3 \(\alpha^{3} \mid \mathrm{A}\)
4 None of these
Explanation:
(C) : Given, \(A\) is square matrix of order 3 and \(\alpha\) is a real number. We know that, if \(A\) is a square matrix of order \(n\) and its determinates if \(|\mathrm{A}|\) And for any scaler \(\mathrm{k},|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|\) Here, \(\mathrm{n}=3\) and \(\mathrm{k}=\alpha\) So, \(|\alpha \mathrm{A}|=\alpha^{3}|\mathrm{~A}|\)
J&K CET-2011
Matrix and Determinant
79104
The value of the determinant \(\left|\begin{array}{lll}4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3}\end{array}\right|=\) is
79106
If \(M\) and \(N\) are square matrices of order 3 where \(\operatorname{det}(M)=2\) and \(\operatorname{det}(N)=3\), then \(\operatorname{det}(3 \mathrm{MN})\) is
1 27
2 81
3 162
4 324
5 121
Explanation:
(C) : We know that, \(|\mathrm{MN}|=|\mathrm{M}||\mathrm{N}|=2 \times 3=6\) Also, for a square matrix of order 3 \(|\mathrm{KA}|=\mathrm{K}^{3}|\mathrm{~A}|\) because each element of the matrix \(\mathrm{A}\) is multiplied by \(\mathrm{k}\) and hence, \((3 \mathrm{MN})=(3)^{3}|\mathrm{MN}|\) \(=27 \times 6\) \(|3 \mathrm{MN}|=162\)
79103
If \(A\) is a square matrix of order 3 and \(\alpha\) is a real number, then determinant \([\alpha \mathrm{A}]\) is equal to
1 \(\alpha^{2}|\mathrm{~A}|\)
2 \(\alpha|\mathrm{A}|\)
3 \(\alpha^{3} \mid \mathrm{A}\)
4 None of these
Explanation:
(C) : Given, \(A\) is square matrix of order 3 and \(\alpha\) is a real number. We know that, if \(A\) is a square matrix of order \(n\) and its determinates if \(|\mathrm{A}|\) And for any scaler \(\mathrm{k},|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|\) Here, \(\mathrm{n}=3\) and \(\mathrm{k}=\alpha\) So, \(|\alpha \mathrm{A}|=\alpha^{3}|\mathrm{~A}|\)
J&K CET-2011
Matrix and Determinant
79104
The value of the determinant \(\left|\begin{array}{lll}4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3}\end{array}\right|=\) is
79106
If \(M\) and \(N\) are square matrices of order 3 where \(\operatorname{det}(M)=2\) and \(\operatorname{det}(N)=3\), then \(\operatorname{det}(3 \mathrm{MN})\) is
1 27
2 81
3 162
4 324
5 121
Explanation:
(C) : We know that, \(|\mathrm{MN}|=|\mathrm{M}||\mathrm{N}|=2 \times 3=6\) Also, for a square matrix of order 3 \(|\mathrm{KA}|=\mathrm{K}^{3}|\mathrm{~A}|\) because each element of the matrix \(\mathrm{A}\) is multiplied by \(\mathrm{k}\) and hence, \((3 \mathrm{MN})=(3)^{3}|\mathrm{MN}|\) \(=27 \times 6\) \(|3 \mathrm{MN}|=162\)
