(B) : It is given that, \(\left|\begin{array}{lll} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{array}\right|\) Applying \(\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}+\mathrm{C}_{2}\) \(\left|\begin{array}{ccc} 1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c \end{array}\right|\) \(=(a+b+c)\left|\begin{array}{lll} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \end{array}\right|\) \(=(a+b+c) \times 0 \quad\left(\because C_{1} \text { and } C_{2} \text { are equal }\right)\) \(=0\)
Kerala CEE-2018
Matrix and Determinant
78984
Let \(A\) be a square matrix, all of whose entries are integers, Then, which one of the following is true?
1 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) need not exist
2 If \(\operatorname{det}(A)= \pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers
3 If \(\operatorname{det}(\mathrm{A}) \neq \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are not-integers
4 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are integers
Explanation:
(D) : Here, matrix \(A\) has all integer elements, its co-factors will also be an integers, so adj (A) will also have integer elements. So, If \(|A|= \pm 1\) We have, \(\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})= \pm 1 \times \operatorname{adj}(\mathrm{A})\) Hence, \(\mathrm{A}^{-1}\) exists, and all the elements are integers.
BCECE-2014
Matrix and Determinant
78991
Let \(A\) be a \(3 \times 3\) matrix and \(B\) be its adj matrix. If \(|B|=64\), then \(|A|\) is equal to
(B) : It is given that, \(\left|\begin{array}{lll} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{array}\right|\) Applying \(\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}+\mathrm{C}_{2}\) \(\left|\begin{array}{ccc} 1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c \end{array}\right|\) \(=(a+b+c)\left|\begin{array}{lll} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \end{array}\right|\) \(=(a+b+c) \times 0 \quad\left(\because C_{1} \text { and } C_{2} \text { are equal }\right)\) \(=0\)
Kerala CEE-2018
Matrix and Determinant
78984
Let \(A\) be a square matrix, all of whose entries are integers, Then, which one of the following is true?
1 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) need not exist
2 If \(\operatorname{det}(A)= \pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers
3 If \(\operatorname{det}(\mathrm{A}) \neq \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are not-integers
4 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are integers
Explanation:
(D) : Here, matrix \(A\) has all integer elements, its co-factors will also be an integers, so adj (A) will also have integer elements. So, If \(|A|= \pm 1\) We have, \(\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})= \pm 1 \times \operatorname{adj}(\mathrm{A})\) Hence, \(\mathrm{A}^{-1}\) exists, and all the elements are integers.
BCECE-2014
Matrix and Determinant
78991
Let \(A\) be a \(3 \times 3\) matrix and \(B\) be its adj matrix. If \(|B|=64\), then \(|A|\) is equal to
(B) : It is given that, \(\left|\begin{array}{lll} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{array}\right|\) Applying \(\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}+\mathrm{C}_{2}\) \(\left|\begin{array}{ccc} 1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c \end{array}\right|\) \(=(a+b+c)\left|\begin{array}{lll} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \end{array}\right|\) \(=(a+b+c) \times 0 \quad\left(\because C_{1} \text { and } C_{2} \text { are equal }\right)\) \(=0\)
Kerala CEE-2018
Matrix and Determinant
78984
Let \(A\) be a square matrix, all of whose entries are integers, Then, which one of the following is true?
1 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) need not exist
2 If \(\operatorname{det}(A)= \pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers
3 If \(\operatorname{det}(\mathrm{A}) \neq \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are not-integers
4 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are integers
Explanation:
(D) : Here, matrix \(A\) has all integer elements, its co-factors will also be an integers, so adj (A) will also have integer elements. So, If \(|A|= \pm 1\) We have, \(\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})= \pm 1 \times \operatorname{adj}(\mathrm{A})\) Hence, \(\mathrm{A}^{-1}\) exists, and all the elements are integers.
BCECE-2014
Matrix and Determinant
78991
Let \(A\) be a \(3 \times 3\) matrix and \(B\) be its adj matrix. If \(|B|=64\), then \(|A|\) is equal to
(B) : It is given that, \(\left|\begin{array}{lll} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{array}\right|\) Applying \(\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}+\mathrm{C}_{2}\) \(\left|\begin{array}{ccc} 1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c \end{array}\right|\) \(=(a+b+c)\left|\begin{array}{lll} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \end{array}\right|\) \(=(a+b+c) \times 0 \quad\left(\because C_{1} \text { and } C_{2} \text { are equal }\right)\) \(=0\)
Kerala CEE-2018
Matrix and Determinant
78984
Let \(A\) be a square matrix, all of whose entries are integers, Then, which one of the following is true?
1 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) need not exist
2 If \(\operatorname{det}(A)= \pm 1\), then \(A^{-1}\) exists but all its entries are not necessarily integers
3 If \(\operatorname{det}(\mathrm{A}) \neq \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are not-integers
4 If \(\operatorname{det}(\mathrm{A})= \pm 1\), then \(\mathrm{A}^{-1}\) exists and all its entries are integers
Explanation:
(D) : Here, matrix \(A\) has all integer elements, its co-factors will also be an integers, so adj (A) will also have integer elements. So, If \(|A|= \pm 1\) We have, \(\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})= \pm 1 \times \operatorname{adj}(\mathrm{A})\) Hence, \(\mathrm{A}^{-1}\) exists, and all the elements are integers.
BCECE-2014
Matrix and Determinant
78991
Let \(A\) be a \(3 \times 3\) matrix and \(B\) be its adj matrix. If \(|B|=64\), then \(|A|\) is equal to
