Matrices Test

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Matrices Test - 20

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Weekly Quiz Competition

Question 1
1 / -0

If $$A=\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$ and $$B=\begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 4 \end{bmatrix}$$, then $$AB$$ equal to

A
$$\begin{bmatrix} 8 & 15 & 16 \\ 5 & 9 & 10 \end{bmatrix}$$

B
$$\begin{bmatrix} 8 & 5 \\ 15 & 9 \\ 16 & 10 \end{bmatrix}$$

C
$$\begin{bmatrix} 8 & 5 \\ 15 & 9 \end{bmatrix}$$

D
None of these

Solution

$$AB=\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}\begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 4 \end{bmatrix}=\begin{bmatrix} 2.1+2.2 & 2.3+3.3 & 2.2+3.4 \\ 1.1+2.2 & 1.3+2.3 & 1.2+2.4 \end{bmatrix}=\begin{bmatrix} 8 & 15 & 16 \\ 5 & 9 & 10 \end{bmatrix}$$

Ans: A
Question 2
1 / -0

For square matrix $$A$$, $$A{A}^{T}$$ is-

A
unit matrix

B
symmetric matrix

C
skew symmetric matrix

D
diagonal matrix

Solution

Since,$${ \left( A{ A }^{ T } \right) }^{ T }={ \left( { A }^{ T } \right) }^{ T }{ A }^{ T }=A{ A }^{ T }$$
Therefore, $$A{ A }^{ T }$$ is symmetric matrix

Ans: B
Question 3
1 / -0

If $$\begin{bmatrix} x & y \\ u & v \end{bmatrix}$$ is symmetric matrix, then

A
$$x+v=0$$

B
$$x-v=0$$

C
$$y+u=0$$

D
$$y-u=0$$

Solution

$$A =\begin{bmatrix} x & y \\ u & v \end{bmatrix}$$
$$A^T =\begin{bmatrix} x & u \\ y & v \end{bmatrix}$$
So for $$A$$ to be symmetric matrix, $$A=A^T$$
$$\Rightarrow y = u\Rightarrow y-u=0$$
Question 4
1 / -0

If $$A= \displaystyle \begin{bmatrix} 1 \\ 3 \end{bmatrix} $$ and $$B= \displaystyle \begin{bmatrix} -1 \\ 4 \end{bmatrix} , $$ then $$A-B =$$

A
$$\displaystyle \begin{bmatrix} 2 \\ -1 \end{bmatrix} $$

B
$$\displaystyle \begin{bmatrix} 10 \\ 1 \end{bmatrix} $$

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

D
$$\displaystyle \begin{bmatrix} 1 \\ 9 \end{bmatrix} $$

Solution

Since, $$A= \displaystyle \begin{bmatrix} 1 \\ 3 \end{bmatrix} $$ $$B = \displaystyle \begin{bmatrix} -1 \\ 4 \end{bmatrix} $$

Therefore, $$A-B =\begin{bmatrix} 2 \\ -1 \end{bmatrix}$$
Question 5
1 / -0

If $$\displaystyle \begin{bmatrix} 2 & 3 \\ 4 & 4 \end{bmatrix} $$+$$\displaystyle \begin{bmatrix} x & 3 \\ y & 1 \end{bmatrix} $$=$$\displaystyle \begin{bmatrix} 10 & 6 \\ 8 & 5 \end{bmatrix},$$ then $$(x,y)=$$

A
$$(4,8)$$

B
$$(8,4)$$

C
$$(1,2)$$

D
$$(2,4)$$

Solution

By equating given matrices, we have
$$2+x=10 \Rightarrow x=8$$
Also, $$4+y=8 \Rightarrow y=4$$.
Question 6
1 / -0

If $$\displaystyle 2\begin{bmatrix}3 &4 \\5 &x \end{bmatrix}+\begin{bmatrix}1 &y \\0 &2 \end{bmatrix}=\begin{bmatrix}7 &0 \\10 &5 \end{bmatrix}$$, then the values of $$x\ and\ y$$ are :

A
$$\displaystyle x=0,y=-2$$

B
$$\displaystyle x=\dfrac{3}{2},y=-8$$

C
$$\displaystyle x=-2,y=-8$$

D
$$\displaystyle x=2,y=8$$

Solution

$$2\left[ \begin{matrix} 3 & 4 \\ 5 & x \end{matrix} \right] +\left[ \begin{matrix} 1 & y \\ 0 & 2 \end{matrix} \right] =\left[ \begin{matrix} 7 & 0 \\ 10 & 5 \end{matrix} \right] $$

$$\Rightarrow \left[ \begin{matrix} 6 & 8 \\ 10 & 2x \end{matrix} \right] +\left[ \begin{matrix} 1 & y \\ 0 & 2 \end{matrix} \right] =\left[ \begin{matrix} 7 & 0 \\ 10 & 5 \end{matrix} \right] $$

$$\Rightarrow \left[ \begin{matrix} 7 & 8+y \\ 10 & 2x+2 \end{matrix} \right] =\left[ \begin{matrix} 7 & 0 \\ 10 & 5 \end{matrix} \right] $$

$$\Rightarrow 8+y=0\;\&\;2x+2=5$$

$$\Rightarrow y=-8\;\&\;2x=3$$
$$\Rightarrow x=\dfrac { 3 }{ 2 } $$

$$\therefore x=\dfrac { 3 }{ 2 }\; \&\;y=-8$$
Hence, the answer is $$x=\dfrac { 3 }{ 2 }\; \&\;y=-8.$$
Question 7
1 / -0

If $$A= \displaystyle \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} $$ and B=$$\displaystyle \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, $$ then $$A+B=$$

A
$$A$$

B
$$B$$

C
$$\displaystyle \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix} $$

D
$$\displaystyle \begin{bmatrix} 0 & 2 \\ 2 & 2 \end{bmatrix} $$

Solution

Given, $$A= \displaystyle \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} $$ and B=$$\displaystyle \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$
Then, $$A+ B = \displaystyle \begin{vmatrix} 2 & 0 \\ 1 & 1 \end{vmatrix}$$
Question 8
1 / -0

If $$P=\displaystyle \begin{bmatrix} 4 & 3 &2 \end{bmatrix} $$ and $$Q = \displaystyle \begin{bmatrix} -1 & 2 &3 \end{bmatrix} ,$$ then $$P-Q=$$

A
$$\displaystyle \begin{bmatrix} 5 & 1 &-1 \end{bmatrix} $$

B
$$\displaystyle \begin{bmatrix} 5 & -1 &-4 \end{bmatrix} $$

C
$$\displaystyle \begin{bmatrix} 5 & 1 &1 \end{bmatrix} $$

D
$$\displaystyle \begin{bmatrix} -4 & -1 &6 \end{bmatrix} $$

Solution

Conditions for the subtraction of matrices:
1. Two matrices should be of the same order (number of rows=number of columns).
2. Subtract the corresponding element of other matrices.
Given, $$P=\displaystyle \begin{bmatrix} 4 & 3 &2 \end{bmatrix} $$ and
$$Q= \displaystyle \begin{bmatrix} -1 & 2 &3 \end{bmatrix} $$
Therefore,
$$P-Q=[4\ 3\ 2] - [-1\ 2\ 3]= [5\ 1\ -1]$$
Question 9
1 / -0

If A is any square matrix, then $$(A\, +\, A^T)$$ is a ............ matrix

A
symmetric

B
skew symmetric

C
scalar

D
identity

Solution

Lets matrix $$A=\begin{bmatrix} a & b & c\\ d & f & g\\ e & h & i\end{bmatrix}$$

Then, matrix $$A^T$$, after transforming rows with each other will be,
$$A^T=\begin{bmatrix} a & d & e\\ b & f & h\\ c & g & i\end{bmatrix}$$

on adding $$A+A^T$$, we get

$$\begin{bmatrix} 2a & (b+d) & (c+e)\\ (b+d) & 2f & (h+g)\\ (e+c) & (h+g) & 2i\end{bmatrix}$$

which is clearly symmetry about its diagonal.
Question 10
1 / -0

Two matrices $$A$$ and $$B$$ are added if

A
both are rectangular

B
both have same order

C
no of columns of A is equal to columns of B

D
no of rows of A is equal to no of columns of B

Solution

While adding two matrices we add the numbers which belong to some row and column of each matrix or two matrices can be added if there are equal number of rows and columns in both.
Hence, both matrices should have same order.
$$\therefore$$ option B is correct

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Matrices Test - 20

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Matrices Test - 20

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Question 1

$$\begin{bmatrix} 8 & 15 & 16 \\ 5 & 9 & 10 \end{bmatrix}$$

$$\begin{bmatrix} 8 & 5 \\ 15 & 9 \\ 16 & 10 \end{bmatrix}$$

$$\begin{bmatrix} 8 & 5 \\ 15 & 9 \end{bmatrix}$$

None of these

Solution

Question 2

unit matrix

symmetric matrix

skew symmetric matrix

diagonal matrix

Solution

Question 3

$$x+v=0$$

$$x-v=0$$

$$y+u=0$$

$$y-u=0$$

Solution

Question 4

$$\displaystyle \begin{bmatrix} 2 \\ -1 \end{bmatrix} $$

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

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

$$\displaystyle \begin{bmatrix} 1 \\ 9 \end{bmatrix} $$

Solution

Question 5

$$(4,8)$$

$$(8,4)$$

$$(1,2)$$

$$(2,4)$$

Solution

Question 6

$$\displaystyle x=0,y=-2$$

$$\displaystyle x=\dfrac{3}{2},y=-8$$

$$\displaystyle x=-2,y=-8$$

$$\displaystyle x=2,y=8$$

Solution

Question 7

$$A$$

$$B$$

$$\displaystyle \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix} $$

$$\displaystyle \begin{bmatrix} 0 & 2 \\ 2 & 2 \end{bmatrix} $$

Solution

Question 8

$$\displaystyle \begin{bmatrix} 5 & 1 &-1 \end{bmatrix} $$

$$\displaystyle \begin{bmatrix} 5 & -1 &-4 \end{bmatrix} $$

$$\displaystyle \begin{bmatrix} 5 & 1 &1 \end{bmatrix} $$

$$\displaystyle \begin{bmatrix} -4 & -1 &6 \end{bmatrix} $$

Solution

Question 9

symmetric

skew symmetric

scalar

identity

Solution

Question 10

both are rectangular

both have same order

no of columns of A is equal to columns of B

no of rows of A is equal to no of columns of B

Solution

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