Matrices Test

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Matrices Test - 21

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Weekly Quiz Competition

Question 1
1 / -0

If $$A$$ and $$B$$ are symmetric matrices of order $$\displaystyle n,\left( A\neq B \right) $$, then

A
$$A+B$$ is skew-symmetric

B
$$A+B$$ is symmetric

C
$$A+B$$ is a diagonal matrix

D
$$A+B$$ is a zero matrix

Solution

Since $$A$$ and $$B$$ are symmetric matrices, $$A' = A$$ and $$B'=B$$
$$\therefore (A+B)'=A'+B'=A+B$$
$$\therefore A+B$$ is symmetric matrix
Question 2
1 / -0

If $$ A= \begin{bmatrix} 1 & 2\end{bmatrix}, B=\begin{bmatrix} 3 & 4\end{bmatrix}$$ then $$A+B=$$

A
$$\begin{bmatrix}1 & 4\end{bmatrix}$$

B
$$\begin{bmatrix}4 & 4\end{bmatrix}$$

C
$$\begin{bmatrix}4 & 6\end{bmatrix}$$

D
None of these

Solution

Given $$A=\begin{bmatrix} 1 & 2 \end{bmatrix}$$
$$B=\begin{bmatrix} 3 & 4 \end{bmatrix}$$
Now, $$A+B=\begin{bmatrix} 1+3 & 2+4 \end{bmatrix}$$
$$\Rightarrow A+B=\begin{bmatrix} 4 & 6 \end{bmatrix}$$
$$\therefore $$Option C is correct
Question 3
1 / -0

Find the output order for the following matrix multiplication $$A_{4 \times 2}\times B_{2\times4}$$?

A
$$2 \times 4$$

B
$$4 \times 4$$

C
$$4 \times2$$

D
Multiplication not possible

Solution

Note:
1. $$m×n$$ is the order of matrix $$AB$$ if the order of matrix $$A$$ is $$m×p$$ and the order of $$B$$ is $$p×n$$.
2. For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix.

Here $$A,B$$ are conformant matrices, Multiplication is possible.
Therefore, $$A$$ is of size $$4 \times 2$$, and $$B$$ is of size $$2 \times 4$$, in which case the output is of size $$4 \times 4$$ and is the matrix product of $$A$$ and $$B$$.
Question 4
1 / -0

If the matrices $$A=\begin{bmatrix}2 & 1 & 3 \\4 & 1 & 0\end{bmatrix}$$ and $$B=\begin{bmatrix}1 & -1\\ 0 & 2 \\5 & 0\end{bmatrix}$$, then AB will be

A
$$\begin{bmatrix}17 & 0 \\4 & -2\end{bmatrix}$$

B
$$\begin{bmatrix}4 & 0 \\ 0 & 4\end{bmatrix}$$

C
$$\begin{bmatrix}17 & 4 \\0 & -2\end{bmatrix}$$

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

Solution

Given, $$A=\begin{bmatrix}2 & 1 & 3 \\ 4 & 1 & 0\end{bmatrix}$$ and $$B=\begin{bmatrix}1 & -1 \\ 0 & 2 \\ 5 & 0\end{bmatrix}$$
Now, $$AB=\begin{bmatrix}2\times1+1\times0+3\times5 & 2\times(-1)+1\times2+3\times0 \\ 4\times1+1\times0+0\times5 & 4\times(-1)+1\times2+0\times0\end{bmatrix}$$
$$=\begin{bmatrix}17 & 0 \\ 4 & -2\end{bmatrix}$$
Question 5
1 / -0

What is the output for the following matrix multiplication $$A_{3 \times 2}\times B_{2\times 3}$$?

A
$$(AB)_{3\times 2}$$

B
$$(AB)_{3\times 3}$$

C
$$(AB)_{2\times 3}$$

D
$$(AB)_{2\times 2}$$

Solution

Here $$A,B$$ are conformant matrices.
Therefore, $$A$$ is of size $$3 \times 2$$, and $$B$$ is of size $$2 \times 3$$, in which case the output is of size $$3 \times 3$$ and is the matrix product of $$A$$ and $$B$$.
The result will be $$(AB)_{3\times 3}.$$
Question 6
1 / -0

Find the output order for the following matrix multiplication $$X_{5 \times 3}\times Y_{3\times 5}$$?

A
$$5 \times 5$$

B
$$3 \times 5$$

C
$$5 \times 3$$

D
$$3 \times 3$$

Solution

Since, $$X$$ is of size $$5 \times 3$$, and $$Y$$ is of size $$3 \times 5$$
$$\therefore$$ The matrix product of $$X$$ and $$Y$$ has order $$5 \times 5.$$
Question 7
1 / -0

Find the value in place of question mark in the following:
$$A_{6 \times 2}\times B_{2\times 6} = C_{?\times6}$$?

A
$$2$$

B
$$6$$

C
$$12$$

D
None of these

Solution

$$A_{6 \times 2}$$ denotes a matrix having $$6$$ rows and $$2$$ columns.
Similarly, $$B_{2 \times 6}$$ denotes a matrix having $$2$$ rows and $$6$$ columns.
When multiplied, we get $$6$$ rows and $$6$$ columns based on the matrix multiplication rule.
Thus, if the product of matrices $$A$$ and $$B$$ is $$C$$, we can write the answer as $$C_{6 \times 6}$$
Question 8
1 / -0

$$A=\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$$ and $$AB=BA=I$$, then B is equal to

A
$$\begin{bmatrix} -\cos\theta & \sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$$

B
$$\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}$$

C
$$\begin{bmatrix} -\sin\theta & \cos\theta \\ \cos\theta & \sin\theta\end{bmatrix}$$

D
$$\begin{bmatrix} \sin\theta & -\cos\theta \\ -\cos\theta & \sin\theta\end{bmatrix}$$

Solution

Given, $$A=\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$ and $$AB=BA=I$$
$$\Rightarrow B=A^{-1}I=A^{-1}$$
$$=\displaystyle\frac{1}{\cos^2\theta +\sin^2\theta}\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}$$
$$\Rightarrow B=\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}$$
Question 9
1 / -0

If $$A=\begin{bmatrix} 3 & x-1 \\ 2x+3 & x+2 \end{bmatrix}$$ is symmetric matrix then the value of $$x$$ is

A
$$4$$

B
$$3$$

C
$$-4$$

D
$$-3$$

Solution

Since $$A$$ is symmetric matrix, then
$$A=A'$$
$$\Rightarrow$$ $$\begin{bmatrix} 3 & x-1 \\ 2x+3 & x+2 \end{bmatrix}=\begin{bmatrix} 3 & 2x+3 \\ x-1 & x+2 \end{bmatrix}$$
$$\Rightarrow $$ $$x-1=2x+3$$ $$\Rightarrow$$ $$x=-4$$
Question 10
1 / -0

What is the output order for the following matrix multiplication $$A_{2 \times 1}\times B_{1\times 2}$$?

A
$$1 \times 1$$

B
$$2 \times 3$$

C
$$2 \times 2$$

D
$$2 \times 1$$

Solution

We know that $$m\times n $$ is the order of matrix $$AB$$ if the order of matrix $$A$$ is $$m\times p$$ and the order of $$B$$ is $$p\times n.$$
Since, $$A$$ is of size $$2 \times 1$$, and $$B$$ is of size $$1 \times 2$$, in which case the output is of size $$2 \times 2$$
and is the matrix product of $$A$$ and $$B$$.

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Matrices Test - 21

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Matrices Test - 21

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Question 1

$$A+B$$ is skew-symmetric

$$A+B$$ is symmetric

$$A+B$$ is a diagonal matrix

$$A+B$$ is a zero matrix

Solution

Question 2

$$\begin{bmatrix}1 & 4\end{bmatrix}$$

$$\begin{bmatrix}4 & 4\end{bmatrix}$$

$$\begin{bmatrix}4 & 6\end{bmatrix}$$

None of these

Solution

Question 3

$$2 \times 4$$

$$4 \times 4$$

$$4 \times2$$

Multiplication not possible

Solution

Question 4

$$\begin{bmatrix}17 & 0 \\4 & -2\end{bmatrix}$$

$$\begin{bmatrix}4 & 0 \\ 0 & 4\end{bmatrix}$$

$$\begin{bmatrix}17 & 4 \\0 & -2\end{bmatrix}$$

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

Solution

Question 5

$$(AB)_{3\times 2}$$

$$(AB)_{3\times 3}$$

$$(AB)_{2\times 3}$$

$$(AB)_{2\times 2}$$

Solution

Question 6

$$5 \times 5$$

$$3 \times 5$$

$$5 \times 3$$

$$3 \times 3$$

Solution

Question 7

$$2$$

$$6$$

$$12$$

None of these

Solution

Question 8

$$\begin{bmatrix} -\cos\theta & \sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$$

$$\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}$$

$$\begin{bmatrix} -\sin\theta & \cos\theta \\ \cos\theta & \sin\theta\end{bmatrix}$$

$$\begin{bmatrix} \sin\theta & -\cos\theta \\ -\cos\theta & \sin\theta\end{bmatrix}$$

Solution

Question 9

$$4$$

$$3$$

$$-4$$

$$-3$$

Solution

Question 10

$$1 \times 1$$

$$2 \times 3$$

$$2 \times 2$$

$$2 \times 1$$

Solution

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