QBManager

Matrix and Determinant

79103 If $A$ is a square matrix of order 3 and $α$ is a real number, then determinant $[α A]$ is equal to

1

α^{2} | A |

2

α | A |

3

α^{3} ∣ A

4 None of these

Explanation:

(C) : Given,
$A$ is square matrix of order 3 and $α$ is a real number.
We know that, if $A$ is a square matrix of order $n$ and its determinates if $| A |$
And for any scaler $k, | kA | = k^{n} | A |$
Here, $n = 3$ and $k = α$
So, $| α A | = α^{3} | A |$

J&K CET-2011

Matrix and Determinant

79104 The value of the determinant $| \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} | =$ is

1 52

2 -24

3 24

4 48

5 -48

Explanation:

(E) : Given, $A = | \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} |$
$| \begin{array}{ccc} 4 & 16 & 64 \\ 3 & 9 & 27 \\ 2 & 4 & 8 \end{array} |$
$| A | = 4 (72 - 10$
$| A | = 4 (72 - 108) - 16 (24 - 54) + 64 (12 - 18)$
$= 4 \times (- 36) - 16 \times (- 30) + 64 \times (- 6)$
$= 4 \times 6 \times 2 [- 3 + 10 - 8]$
$| A | = - 48$

Kerala CEE-2021

Matrix and Determinant

79105 If $| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$ , then the values of $x$ are

1

5, - 3

2 5,3

3

- 5, 3

4 2,3

5

- 2, - 3

Explanation:

(A) : Given,
$| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$
$1 (x^{2} - 3) - 2 (0 + 6) + 1 (0 - 2 x) = 0$
$x^{2} - 3 - 12 - 2 x = 0$
$x^{2} - 2 x - 15 = 0$
$x^{2} - 5 x + 3 x - 15 = 0$
$x (x - 5) + 3 (x - 5) = 0$
$(x + 3) (x - 5) = 0$
$x = - 3, 5$

Kerala CEE-2021

Matrix and Determinant

79106 If $M$ and $N$ are square matrices of order 3 where $\det (M) = 2$ and $\det (N) = 3$ , then $\det (3 MN)$ is

1 27

2 81

3 162

4 324

5 121

Explanation:

(C) : We know that,
$| MN | = | M | | N | = 2 \times 3 = 6$
Also, for a square matrix of order 3
$| KA | = K^{3} | A |$ because each element of the matrix $A$ is multiplied by $k$ and hence,
$(3 MN) = (3)^{3} | MN |$
$= 27 \times 6$
$| 3 MN | = 162$

Kerala CEE-2019

Matrix and Determinant

79103 If $A$ is a square matrix of order 3 and $α$ is a real number, then determinant $[α A]$ is equal to

1

α^{2} | A |

2

α | A |

3

α^{3} ∣ A

4 None of these

Explanation:

(C) : Given,
$A$ is square matrix of order 3 and $α$ is a real number.
We know that, if $A$ is a square matrix of order $n$ and its determinates if $| A |$
And for any scaler $k, | kA | = k^{n} | A |$
Here, $n = 3$ and $k = α$
So, $| α A | = α^{3} | A |$

J&K CET-2011

Matrix and Determinant

79104 The value of the determinant $| \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} | =$ is

1 52

2 -24

3 24

4 48

5 -48

Explanation:

(E) : Given, $A = | \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} |$
$| \begin{array}{ccc} 4 & 16 & 64 \\ 3 & 9 & 27 \\ 2 & 4 & 8 \end{array} |$
$| A | = 4 (72 - 10$
$| A | = 4 (72 - 108) - 16 (24 - 54) + 64 (12 - 18)$
$= 4 \times (- 36) - 16 \times (- 30) + 64 \times (- 6)$
$= 4 \times 6 \times 2 [- 3 + 10 - 8]$
$| A | = - 48$

Kerala CEE-2021

Matrix and Determinant

79105 If $| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$ , then the values of $x$ are

1

5, - 3

2 5,3

3

- 5, 3

4 2,3

5

- 2, - 3

Explanation:

(A) : Given,
$| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$
$1 (x^{2} - 3) - 2 (0 + 6) + 1 (0 - 2 x) = 0$
$x^{2} - 3 - 12 - 2 x = 0$
$x^{2} - 2 x - 15 = 0$
$x^{2} - 5 x + 3 x - 15 = 0$
$x (x - 5) + 3 (x - 5) = 0$
$(x + 3) (x - 5) = 0$
$x = - 3, 5$

Kerala CEE-2021

Matrix and Determinant

79106 If $M$ and $N$ are square matrices of order 3 where $\det (M) = 2$ and $\det (N) = 3$ , then $\det (3 MN)$ is

1 27

2 81

3 162

4 324

5 121

Explanation:

(C) : We know that,
$| MN | = | M | | N | = 2 \times 3 = 6$
Also, for a square matrix of order 3
$| KA | = K^{3} | A |$ because each element of the matrix $A$ is multiplied by $k$ and hence,
$(3 MN) = (3)^{3} | MN |$
$= 27 \times 6$
$| 3 MN | = 162$

Kerala CEE-2019

Matrix and Determinant

79103 If $A$ is a square matrix of order 3 and $α$ is a real number, then determinant $[α A]$ is equal to

1

α^{2} | A |

2

α | A |

3

α^{3} ∣ A

4 None of these

Explanation:

(C) : Given,
$A$ is square matrix of order 3 and $α$ is a real number.
We know that, if $A$ is a square matrix of order $n$ and its determinates if $| A |$
And for any scaler $k, | kA | = k^{n} | A |$
Here, $n = 3$ and $k = α$
So, $| α A | = α^{3} | A |$

J&K CET-2011

Matrix and Determinant

79104 The value of the determinant $| \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} | =$ is

1 52

2 -24

3 24

4 48

5 -48

Explanation:

(E) : Given, $A = | \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} |$
$| \begin{array}{ccc} 4 & 16 & 64 \\ 3 & 9 & 27 \\ 2 & 4 & 8 \end{array} |$
$| A | = 4 (72 - 10$
$| A | = 4 (72 - 108) - 16 (24 - 54) + 64 (12 - 18)$
$= 4 \times (- 36) - 16 \times (- 30) + 64 \times (- 6)$
$= 4 \times 6 \times 2 [- 3 + 10 - 8]$
$| A | = - 48$

Kerala CEE-2021

Matrix and Determinant

79105 If $| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$ , then the values of $x$ are

1

5, - 3

2 5,3

3

- 5, 3

4 2,3

5

- 2, - 3

Explanation:

(A) : Given,
$| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$
$1 (x^{2} - 3) - 2 (0 + 6) + 1 (0 - 2 x) = 0$
$x^{2} - 3 - 12 - 2 x = 0$
$x^{2} - 2 x - 15 = 0$
$x^{2} - 5 x + 3 x - 15 = 0$
$x (x - 5) + 3 (x - 5) = 0$
$(x + 3) (x - 5) = 0$
$x = - 3, 5$

Kerala CEE-2021

Matrix and Determinant

79106 If $M$ and $N$ are square matrices of order 3 where $\det (M) = 2$ and $\det (N) = 3$ , then $\det (3 MN)$ is

1 27

2 81

3 162

4 324

5 121

Explanation:

(C) : We know that,
$| MN | = | M | | N | = 2 \times 3 = 6$
Also, for a square matrix of order 3
$| KA | = K^{3} | A |$ because each element of the matrix $A$ is multiplied by $k$ and hence,
$(3 MN) = (3)^{3} | MN |$
$= 27 \times 6$
$| 3 MN | = 162$

Kerala CEE-2019

Matrix and Determinant

79103 If $A$ is a square matrix of order 3 and $α$ is a real number, then determinant $[α A]$ is equal to

1

α^{2} | A |

2

α | A |

3

α^{3} ∣ A

4 None of these

Explanation:

(C) : Given,
$A$ is square matrix of order 3 and $α$ is a real number.
We know that, if $A$ is a square matrix of order $n$ and its determinates if $| A |$
And for any scaler $k, | kA | = k^{n} | A |$
Here, $n = 3$ and $k = α$
So, $| α A | = α^{3} | A |$

J&K CET-2011

Matrix and Determinant

79104 The value of the determinant $| \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} | =$ is

1 52

2 -24

3 24

4 48

5 -48

Explanation:

(E) : Given, $A = | \begin{array}{lll} 4 & 4^{2} & 4^{3} \\ 3 & 3^{2} & 3^{3} \\ 2 & 2^{2} & 2^{3} \end{array} |$
$| \begin{array}{ccc} 4 & 16 & 64 \\ 3 & 9 & 27 \\ 2 & 4 & 8 \end{array} |$
$| A | = 4 (72 - 10$
$| A | = 4 (72 - 108) - 16 (24 - 54) + 64 (12 - 18)$
$= 4 \times (- 36) - 16 \times (- 30) + 64 \times (- 6)$
$= 4 \times 6 \times 2 [- 3 + 10 - 8]$
$| A | = - 48$

Kerala CEE-2021

Matrix and Determinant

79105 If $| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$ , then the values of $x$ are

1

5, - 3

2 5,3

3

- 5, 3

4 2,3

5

- 2, - 3

Explanation:

(A) : Given,
$| \begin{array}{ccc} 1 & 2 & 1 \\ 0 & x & - 3 \\ 2 & - 1 & x \end{array} | = 0$
$1 (x^{2} - 3) - 2 (0 + 6) + 1 (0 - 2 x) = 0$
$x^{2} - 3 - 12 - 2 x = 0$
$x^{2} - 2 x - 15 = 0$
$x^{2} - 5 x + 3 x - 15 = 0$
$x (x - 5) + 3 (x - 5) = 0$
$(x + 3) (x - 5) = 0$
$x = - 3, 5$

Kerala CEE-2021

Matrix and Determinant

79106 If $M$ and $N$ are square matrices of order 3 where $\det (M) = 2$ and $\det (N) = 3$ , then $\det (3 MN)$ is

1 27

2 81

3 162

4 324

5 121

Explanation:

(C) : We know that,
$| MN | = | M | | N | = 2 \times 3 = 6$
Also, for a square matrix of order 3
$| KA | = K^{3} | A |$ because each element of the matrix $A$ is multiplied by $k$ and hence,
$(3 MN) = (3)^{3} | MN |$
$= 27 \times 6$
$| 3 MN | = 162$

Kerala CEE-2019

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