Determinants Test

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Determinants Test - 40

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Weekly Quiz Competition

Question 1
1 / -0

If $$x, y, z$$ are positive numbers, then value of the determinant $$\begin{vmatrix}1 & log_xy & log_xz \\ log_yx & 1 & log_yz\\ log_zx & log_zy & 1\end{vmatrix}$$ is equal to

A
$$0$$

B
$$3$$

C
$$log xyz$$

D
none

Solution

The value of the determinant is $$\begin{vmatrix} 1 & \log _{ x }{ y } & \log _{ x }{ z } \\ \log _{ y }{ x } & 1 & \log _{ y }{ z } \\ \log _{ z }{ x } & \log _{ z }{ y } & 1 \end{vmatrix}$$

$$ =\left( 1-\log _{ y }{ z\log _{ z }{ y } } \right) -\log _{ x }{ y } \left( \log _{ y }{ x } -\log _{ y }{ z\log _{ z }{ x } } \right) +\log _{ x }{ z } \left( \log _{ y }{ x\log _{ z }{ y } -\log _{ z }{ x } } \right)$$ $$[\because log{_n}^m=\dfrac{logm}{logn}]$$

$$ =0-1+1+1-1=0$$
Question 2
1 / -0

If $$\Delta =\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$$ and $$A_2, B_2, C_2$$ are respectively cofactors of $$a_2, b_2, c_2$$ then $$a_1A_2 + b_1B_2 + c_1C_2$$ is equal to

A
$$-\Delta$$

B
0

C
$$\Delta$$

D
none of these

Solution

$$\Delta =\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } & { c }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } & { c }_{ 2 } \\ { a }_{ 3 } & { b }_{ 3 } & { c }_{ 3 } \end{vmatrix}\\ { A }_{ 2 }=-\begin{vmatrix} { b }_{ 1 } & { c }_{ 1 } \\ { b }_{ 3 } & { c }_{ 3 } \end{vmatrix}={ b }_{ 3 }{ c }_{ 1 }-{ b }_{ 1 }{ c }_{ 3 }\\ B_{ 2 }=\begin{vmatrix} { a }_{ 1 } & { c }_{ 1 } \\ { a }_{ 3 } & { c }_{ 3 } \end{vmatrix}={ c }_{ 3 }{ a }_{ 1 }-{ c }_{ 1 }{ a }_{ 3 }\\ C_{ 3 }=-\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } \\ { a }_{ 3 } & { b }_{ 3 } \end{vmatrix}={ a }_{ 3 }{ b }_{ 1 }-{ a }_{ 1 }{ b }_{ 3 }\\ \therefore { a }_{ 1 }{ A }_{ 2 }+{ b }_{ 1 }B_{ 2 }+{ c }_{ 1 }C_{ 3 }={ a }_{ 1 }{ b }_{ 3 }{ c }_{ 1 }-{ a }_{ 1 }{ b }_{ 1 }{ c }_{ 3 }+{ a }_{ 1 }{ b }_{ 1 }{ c }_{ 3 }-{ a }_{ 3 }{ b }_{ 1 }{ c }_{ 1 }+{ a }_{ 3 }{ b }_{ 1 }{ c }_{ 1 }-{ a }_{ 1 }{ b }_{ 3 }{ c }_{ 1 }=0$$
Question 3
1 / -0

If [a] denotes the greatest integer less than or equal to a and $$-1 \leq x < 0, 0 \leq y < 1, 1 \leq z < 2$$, then $$\begin{vmatrix}[x]+1 & [y] & [z] \\ [x] & [y]+1 & [z] \\ [x] & [y] & [z]+1\end{vmatrix}$$ is equal to

A
$$[x]$$

B
$$[y]$$

C
$$[z]$$

D
None of these

Solution

$$- 1 \leq x < 0\rightarrow [x] =- 1$$

$$0 \leq y < 1 \rightarrow [y] = 0$$

$$1 \leq z < 2 \rightarrow [z] = 1$$

$$\begin{vmatrix}0 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & 0 & 2\end{vmatrix}$$

$$=1=[z]$$
Question 4
1 / -0

If cofactor of 2x in the determinant $$\begin{vmatrix}x & 1 & -2 \\ 1 & 2x & x-1 \\ x-1 & x & 0\end{vmatrix}$$ is zero, then $$x$$ equals to

A
0

B
2

C
1

D
-1

Solution

Cofactor of $$2x=0$$
$$\displaystyle \Rightarrow { C }_{ 22 }=0$$
$$\displaystyle \Rightarrow -1^{(2+2)}\begin{vmatrix} x & -2 \\ x-1 & 0 \end{vmatrix}=0$$
$$\displaystyle \Rightarrow x\times 0-\left( -2 \right) \left( x-1 \right) =0$$
$$\displaystyle \Rightarrow 2\left( x-1 \right) =0\Rightarrow x=1$$
Question 5
1 / -0

If $$A =$$$$\displaystyle \begin{bmatrix}\alpha & 2\\ 2 & \alpha \end{bmatrix}$$ and $$\displaystyle \left | A \right |^{3}=125$$ then the value of $$\displaystyle \alpha $$ is

A
$$\displaystyle \pm 1$$

B
$$\displaystyle \pm 2$$

C
$$\displaystyle \pm 3$$

D
$$\displaystyle \pm 5$$

Solution

We know $$\left| { A }^{ n } \right| ={ \left| A \right| }^{ n }$$
Since, $$\left| { A }^{ 3 } \right| =125\Rightarrow { \left| A \right| }^{ 3 }=125$$
$$\Rightarrow \begin{vmatrix} \alpha & 2 \\ 2 & \alpha \end{vmatrix}=5$$
$$\Rightarrow { \alpha }^{ 2 }-4=5$$
$$\Rightarrow \alpha =\pm 3$$
Question 6
1 / -0

If A and B are square matrices of order 3, then

A
$$adj(AB) = adjB.adjA$$

B
$$\displaystyle (A + B)^{-1} = A^{-1} + B^{-1}$$

C
$$\displaystyle AB = 0 \Rightarrow |A| = 0 or |B| = 0$$

D
$$AB \neq 0 \Rightarrow |A| = 0 and |B| = 0$$

Solution

If $$A$$ and $$B$$ are square matrices of order $$3$$,
Then $$Adj(AB)=AdjB. AdjA$$
Where, $$Adj(A)={ A }^{ -1 }\left| A \right| $$
$$Adj(B)={ B }^{ -1 }\left| B \right| $$
$$Adj(AB)={ (AB) }^{ -1 }\left| AB \right| =B^{-1}A^{-1}|AB|=AdjB.AdjA$$
Question 7
1 / -0

Consider the determinant $$\Delta=\begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}$$
$$M_{ij} =$$ Minor of the element of $$i^{th}$$ row & $$j^{th}$$ column.
$$C_{ij} =$$ Cofactor of element of $$i^{th}$$ row & $$j^{th}$$ column.
$$a_3M_{13} - b_3M_{23} + c_3M_{33}$$ is equal to

A
$$0$$

B
$$4\Delta$$

C
$$2\Delta$$

D
$$\Delta$$

Solution

$${ a }_{ 3 }{ M }_{ 13 }-{ b }_{ 2 }{ M }_{ 23 }+{ c }_{ 3 }{ M }_{ 33 }$$

$$ ={ a }_{ 3 }\begin{vmatrix} { b }_{ 1 }\quad & { b }_{ 2 } \\ { c }_{ 1 } & { c }_{ 2 } \end{vmatrix}-{ b }_{ 3 }\begin{vmatrix} { a }_{ 1 }\quad & { a }_{ 2 } \\ { c }_{ 1 } & { c }_{ 2 } \end{vmatrix}+{ c }_{ 3 }\begin{vmatrix} { a }_{ 1 }\quad & { a }_{ 2 } \\ { b }_{ 1 } & { b }_{ 2 } \end{vmatrix}$$

Is equal to the expansion of $$\triangle $$ along $${ C }_{ 3 }$$
Question 8
1 / -0

Consider the determinant $$\Delta=\begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}$$
$$M_{ij} =$$ Minor of the element of $$i^{th}$$ row & $$j^{th}$$ column.
$$C_{ij} =$$ Cofactor of element of $$i^{th}$$ row & $$j^{th}$$ column.
Value of $$b_1.C_{31} + b_2.C_{32} + b_3.C_{33}$$ is

A
$$0$$

B
$$\Delta$$

C
$$2\Delta$$

D
$$\Delta^2$$

Solution

Value of $${ b }_{ 1 }.{ C }_{ 31 }+{ b }_{ 2 }.C_{ 32 }+b_{ 3 }.{ C }_{ 33 }$$

$$={ b }_{ 1 }.{ \left( -1 \right) }^{ 3+1 }\begin{vmatrix} { a }_{ 2 }\quad & { a }_{ 3 } \\ { b }_{ 2 } & { b }_{ 3 } \end{vmatrix}+{ b }_{ 3 }.{ \left( -1 \right) }^{ 3+2 }\begin{vmatrix} { a }_{ 1 }\quad & { a }_{ 3 } \\ { b }_{ 1 } & { b }_{ 3 } \end{vmatrix}+{ b }_{ 3 }.{ \left( -1 \right) }^{ 3+3 }\begin{vmatrix} { a }_{ 1 }\quad & { a }_{ 3 } \\ { b }_{ 1 } & { b }_{ 2 } \end{vmatrix}$$

$$={ b }_{ 1 }\left( { b }_{ 2 }{ a }_{ 3 }-{ a }_{ 2 }{ b }_{ 3 } \right) -{ b }_{ 2 }\left( { b }_{ 1 }{ a }_{ 3 }-{ a }_{ 1 }{ b }_{ 3 } \right) +{ b }_{ 3 }\left( { b }_{ 1 }{ a }_{ 3 }-{ a }_{ 1 }{ b }_{ 2 } \right)$$

$$=0$$
Question 9
1 / -0

If $$A=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$$, then $$A(Adj. A)$$ equals-

A
$$\begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}$$

B
$$\begin{bmatrix} 0 & 10 \\ 10 & 0 \end{bmatrix}$$

C
$$\begin{bmatrix} 10 & 1 \\ 1 & 10 \end{bmatrix}$$

D
none of these

Solution

$$A=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$$
$$adjA=\begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}$$
$$A\left( adjA \right) =\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}=\begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}$$
Question 10
1 / -0

If $$A=\begin{bmatrix} 1 & -2 & 3 \\ 4 & 0 & -1 \\ -3 & 1 & 5 \end{bmatrix}$$, then $${(adj. A)}_{23}$$ is equal to

A
$$13$$

B
$$-13$$

C
$$5$$

D
$$-5$$

Solution

$$A=\begin{bmatrix} 1 & -2 & 3 \\ 4 & 0 & -1 \\ -3 & 1 & 5 \end{bmatrix}$$

$${(adj. A)}_{23}={C}_{32}$$

So cofactor of $${a}_{32}$$

$${C}_{32}={(-1)}^{3+2}(-1-12)=13$$

Ans: A

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Re-Attempt Weekly Quiz Competition

Determinants Test - 40

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Determinants Test - 40

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Question 1

$$0$$

$$3$$

$$log xyz$$

none

Solution

Question 2

$$-\Delta$$

0

$$\Delta$$

none of these

Solution

Question 3

$$[x]$$

$$[y]$$

$$[z]$$

None of these

Solution

Question 4

0

2

1

-1

Solution

Question 5

$$\displaystyle \pm 1$$

$$\displaystyle \pm 2$$

$$\displaystyle \pm 3$$

$$\displaystyle \pm 5$$

Solution

Question 6

$$adj(AB) = adjB.adjA$$

$$\displaystyle (A + B)^{-1} = A^{-1} + B^{-1}$$

$$\displaystyle AB = 0 \Rightarrow |A| = 0 or |B| = 0$$

$$AB \neq 0 \Rightarrow |A| = 0 and |B| = 0$$

Solution

Question 7

$$0$$

$$4\Delta$$

$$2\Delta$$

$$\Delta$$

Solution

Question 8

$$0$$

$$\Delta$$

$$2\Delta$$

$$\Delta^2$$

Solution

Question 9

$$\begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}$$

$$\begin{bmatrix} 0 & 10 \\ 10 & 0 \end{bmatrix}$$

$$\begin{bmatrix} 10 & 1 \\ 1 & 10 \end{bmatrix}$$

none of these

Solution

Question 10

$$13$$

$$-13$$

$$5$$

$$-5$$

Solution

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